Metamath Proof Explorer


Theorem binom2d

Description: Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023)

Ref Expression
Hypotheses binom2d.1
|- ( ph -> A e. CC )
binom2d.2
|- ( ph -> B e. CC )
Assertion binom2d
|- ( ph -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )

Proof

Step Hyp Ref Expression
1 binom2d.1
 |-  ( ph -> A e. CC )
2 binom2d.2
 |-  ( ph -> B e. CC )
3 binom2
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )