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Metamath Proof Explorer

Theorem sqmid3api

Description: Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022)

Ref Expression
Hypotheses sqmid3api.a 
|- A e. CC
sqmid3api.n 
|- N e. CC
sqmid3api.b 
|- ( A + N ) = B
sqmid3api.c 
|- ( B + N ) = C
Assertion sqmid3api 
|- ( B x. B ) = ( ( A x. C ) + ( N x. N ) )

Proof

Step Hyp Ref Expression
1 sqmid3api.a 
 |-  A e. CC
2 sqmid3api.n 
 |-  N e. CC
3 sqmid3api.b 
 |-  ( A + N ) = B
4 sqmid3api.c 
 |-  ( B + N ) = C
5 1 2 1 2 muladdi 
 |-  ( ( A + N ) x. ( A + N ) ) = ( ( ( A x. A ) + ( N x. N ) ) + ( ( A x. N ) + ( A x. N ) ) )
6 3 3 oveq12i 
 |-  ( ( A + N ) x. ( A + N ) ) = ( B x. B )
7 1 1 mulcli 
 |-  ( A x. A ) e. CC
8 2 2 mulcli 
 |-  ( N x. N ) e. CC
9 1 2 mulcli 
 |-  ( A x. N ) e. CC
10 9 9 addcli 
 |-  ( ( A x. N ) + ( A x. N ) ) e. CC
11 7 8 10 add32i 
 |-  ( ( ( A x. A ) + ( N x. N ) ) + ( ( A x. N ) + ( A x. N ) ) ) = ( ( ( A x. A ) + ( ( A x. N ) + ( A x. N ) ) ) + ( N x. N ) )
12 1 2 addcli 
 |-  ( A + N ) e. CC
13 1 12 2 adddii 
 |-  ( A x. ( ( A + N ) + N ) ) = ( ( A x. ( A + N ) ) + ( A x. N ) )
14 3 oveq1i 
 |-  ( ( A + N ) + N ) = ( B + N )
15 14 4 eqtri 
 |-  ( ( A + N ) + N ) = C
16 15 oveq2i 
 |-  ( A x. ( ( A + N ) + N ) ) = ( A x. C )
17 1 1 2 adddii 
 |-  ( A x. ( A + N ) ) = ( ( A x. A ) + ( A x. N ) )
18 17 oveq1i 
 |-  ( ( A x. ( A + N ) ) + ( A x. N ) ) = ( ( ( A x. A ) + ( A x. N ) ) + ( A x. N ) )
19 7 9 9 addassi 
 |-  ( ( ( A x. A ) + ( A x. N ) ) + ( A x. N ) ) = ( ( A x. A ) + ( ( A x. N ) + ( A x. N ) ) )
20 18 19 eqtri 
 |-  ( ( A x. ( A + N ) ) + ( A x. N ) ) = ( ( A x. A ) + ( ( A x. N ) + ( A x. N ) ) )
21 13 16 20 3eqtr3ri 
 |-  ( ( A x. A ) + ( ( A x. N ) + ( A x. N ) ) ) = ( A x. C )
22 21 oveq1i 
 |-  ( ( ( A x. A ) + ( ( A x. N ) + ( A x. N ) ) ) + ( N x. N ) ) = ( ( A x. C ) + ( N x. N ) )
23 11 22 eqtri 
 |-  ( ( ( A x. A ) + ( N x. N ) ) + ( ( A x. N ) + ( A x. N ) ) ) = ( ( A x. C ) + ( N x. N ) )
24 5 6 23 3eqtr3i 
 |-  ( B x. B ) = ( ( A x. C ) + ( N x. N ) )