This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem redivmuld

Description: Relationship between division and multiplication. (Contributed by SN, 25-Nov-2025)

Ref Expression
Hypotheses redivmuld.a 
|- ( ph -> A e. RR )
redivmuld.b 
|- ( ph -> B e. RR )
redivmuld.c 
|- ( ph -> C e. RR )
redivmuld.z 
|- ( ph -> C =/= 0 )
Assertion redivmuld 
|- ( ph -> ( ( A /R C ) = B <-> ( C x. B ) = A ) )

Proof

Step Hyp Ref Expression
1 redivmuld.a 
 |-  ( ph -> A e. RR )
2 redivmuld.b 
 |-  ( ph -> B e. RR )
3 redivmuld.c 
 |-  ( ph -> C e. RR )
4 redivmuld.z 
 |-  ( ph -> C =/= 0 )
5 1 3 4 redivvald 
 |-  ( ph -> ( A /R C ) = ( iota_ x e. RR ( C x. x ) = A ) )
6 5 eqeq1d 
 |-  ( ph -> ( ( A /R C ) = B <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) )
7 1 3 4 rediveud 
 |-  ( ph -> E! x e. RR ( C x. x ) = A )
8 oveq2 
 |-  ( x = B -> ( C x. x ) = ( C x. B ) )
9 8 eqeq1d 
 |-  ( x = B -> ( ( C x. x ) = A <-> ( C x. B ) = A ) )
10 9 riota2 
 |-  ( ( B e. RR /\ E! x e. RR ( C x. x ) = A ) -> ( ( C x. B ) = A <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) )
11 2 7 10 syl2anc 
 |-  ( ph -> ( ( C x. B ) = A <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) )
12 6 11 bitr4d 
 |-  ( ph -> ( ( A /R C ) = B <-> ( C x. B ) = A ) )