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Metamath Proof Explorer

Theorem domnlcanbOLD

Description: Obsolete version of domnlcanb as of 21-Jun-2025. (Contributed by Thierry Arnoux, 8-Jun-2025) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses domncanOLD.b 
|- B = ( Base ` R )
domncanOLD.1 
|- .0. = ( 0g ` R )
domncanOLD.m 
|- .x. = ( .r ` R )
domncanOLD.x 
|- ( ph -> X e. ( B \ { .0. } ) )
domncanOLD.y 
|- ( ph -> Y e. B )
domncanOLD.z 
|- ( ph -> Z e. B )
domnlcanbOLD.r 
|- ( ph -> R e. Domn )
Assertion domnlcanbOLD 
|- ( ph -> ( ( X .x. Y ) = ( X .x. Z ) <-> Y = Z ) )

Proof

Step Hyp Ref Expression
1 domncanOLD.b 
 |-  B = ( Base ` R )
2 domncanOLD.1 
 |-  .0. = ( 0g ` R )
3 domncanOLD.m 
 |-  .x. = ( .r ` R )
4 domncanOLD.x 
 |-  ( ph -> X e. ( B \ { .0. } ) )
5 domncanOLD.y 
 |-  ( ph -> Y e. B )
6 domncanOLD.z 
 |-  ( ph -> Z e. B )
7 domnlcanbOLD.r 
 |-  ( ph -> R e. Domn )
8 4 adantr 
 |-  ( ( ph /\ ( X .x. Y ) = ( X .x. Z ) ) -> X e. ( B \ { .0. } ) )
9 5 adantr 
 |-  ( ( ph /\ ( X .x. Y ) = ( X .x. Z ) ) -> Y e. B )
10 6 adantr 
 |-  ( ( ph /\ ( X .x. Y ) = ( X .x. Z ) ) -> Z e. B )
11 7 adantr 
 |-  ( ( ph /\ ( X .x. Y ) = ( X .x. Z ) ) -> R e. Domn )
12 simpr 
 |-  ( ( ph /\ ( X .x. Y ) = ( X .x. Z ) ) -> ( X .x. Y ) = ( X .x. Z ) )
13 1 2 3 8 9 10 11 12 domnlcan 
 |-  ( ( ph /\ ( X .x. Y ) = ( X .x. Z ) ) -> Y = Z )
14 simpr 
 |-  ( ( ph /\ Y = Z ) -> Y = Z )
15 14 oveq2d 
 |-  ( ( ph /\ Y = Z ) -> ( X .x. Y ) = ( X .x. Z ) )
16 13 15 impbida 
 |-  ( ph -> ( ( X .x. Y ) = ( X .x. Z ) <-> Y = Z ) )