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Metamath Proof Explorer

Theorem muldivdid

Description: Distribution of division over addition with a multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025)

Ref Expression
Hypotheses muldivdid.1 
|- ( ph -> A e. CC )
muldivdid.2 
|- ( ph -> B e. CC )
muldivdid.3 
|- ( ph -> C e. CC )
muldivdid.4 
|- ( ph -> B =/= 0 )
Assertion muldivdid 
|- ( ph -> ( ( ( A x. B ) + C ) / B ) = ( A + ( C / B ) ) )

Proof

Step Hyp Ref Expression
1 muldivdid.1 
 |-  ( ph -> A e. CC )
2 muldivdid.2 
 |-  ( ph -> B e. CC )
3 muldivdid.3 
 |-  ( ph -> C e. CC )
4 muldivdid.4 
 |-  ( ph -> B =/= 0 )
5 1 2 mulcomd 
 |-  ( ph -> ( A x. B ) = ( B x. A ) )
6 5 oveq1d 
 |-  ( ph -> ( ( A x. B ) + C ) = ( ( B x. A ) + C ) )
7 6 oveq1d 
 |-  ( ph -> ( ( ( A x. B ) + C ) / B ) = ( ( ( B x. A ) + C ) / B ) )
8 muldivdir 
 |-  ( ( A e. CC /\ C e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( ( B x. A ) + C ) / B ) = ( A + ( C / B ) ) )
9 1 3 2 4 8 syl112anc 
 |-  ( ph -> ( ( ( B x. A ) + C ) / B ) = ( A + ( C / B ) ) )
10 7 9 eqtrd 
 |-  ( ph -> ( ( ( A x. B ) + C ) / B ) = ( A + ( C / B ) ) )