I wanted to check how far I can get in formalizing this problem and its
solution. But I stuck in the very beginning.
Firstly, I formalized the initial conditions as Mario suggested:
hyp1: |- 0 < k
hyp2: |- 0 < l
hyp3: |- F = ( x e. RR |-> ( ( ( k x. ( x ^ 2 ) ) - ( ( 2 x. k ) x. x ) ) +
l ) )
hyp4: |- { x | ( F ` x ) = 4 } = { a , b }
hyp5: |- ( ( ( a - b ) ^ 2 ) + ( ( ( F ` a ) - ( F ` b ) ) ^ 2 ) ) = ( 6 ^
2 )
hyp6: |- ( ( ( a ^ 2 ) + ( ( F ` a ) ^ 2 ) ) + ( ( b ^ 2 ) + ( ( F ` b ) ^
2 ) ) ) = c
I used { a , b } instead of { A , B } because then I can easily prove |- a
e. _V, which is used in the proof below. Also, as I understand, a and b
represent x-coordinates of the corresponding points.
Next, I wanted to simplify hyp5, by proving that F(a) = F(b) = 4, so the
hyp5 would be |- ( ( a - b ) ^ 2 ) = ( 6 ^ 2 ). But that’s where I am
stuck. I can prove |- [ a / x ] ( F ` x ) = 4 which looks a right direction
to move in:
1| | vex | |- a e. _V
2| 1 | prid1 | |- a e. { a , b }
3| | hyp4 | |- { x | ( F ` x ) = 4 } = { a , b }
4| 3 | eleq2i | |- ( a e. { x | ( F ` x ) = 4 } <-> a e. { a , b } )
5| 2,4 | mpbir | |- a e. { x | ( F ` x ) = 4 }
6| | df-clab | |- ( a e. { x | ( F ` x ) = 4 } <-> [ a / x ] ( F ` x )
= 4 )
7| 5,6 | mpbi | |- [ a / x ] ( F ` x ) = 4
But I still cannot prove |- ( F ` a ) = 4. Any suggestions what approaches
I can try to prove this?
On Sunday, July 28, 2024 at 11:54:47 PM UTC+2 [email protected] wrote:
> On Sun, Jul 28, 2024 at 11:43 PM Glauco <[email protected]> wrote:
>
>> I maybe wrong, but my feeling is that what Jagra calls A , in Mario's
>> translation is actually < A , 4 > (or < A, F(A) > , if you prefer).
>
>
> I meant it to be interpreted as <A, F(A)>, and part of the proof would be
> showing that F(A) = 4 so that the rest of the statement simplifies. (But
> that would seem to be part of the proof, not the formalization of the
> statement, if we want to read it literally.)
>
>
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