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Metamath Proof Explorer

Theorem crng32d

Description: Commutative/associative law that swaps the last two factors in a triple product in a commutative ring. See also mul32d . (Contributed by Thierry Arnoux, 4-May-2025)

Ref Expression
Hypotheses crng32d.b 
|- B = ( Base ` R )
crng32d.t 
|- .x. = ( .r ` R )
crng32d.r 
|- ( ph -> R e. CRing )
crng32d.x 
|- ( ph -> X e. B )
crng32d.y 
|- ( ph -> Y e. B )
crng32d.z 
|- ( ph -> Z e. B )
Assertion crng32d 
|- ( ph -> ( ( X .x. Y ) .x. Z ) = ( ( X .x. Z ) .x. Y ) )

Proof

Step Hyp Ref Expression
1 crng32d.b 
 |-  B = ( Base ` R )
2 crng32d.t 
 |-  .x. = ( .r ` R )
3 crng32d.r 
 |-  ( ph -> R e. CRing )
4 crng32d.x 
 |-  ( ph -> X e. B )
5 crng32d.y 
 |-  ( ph -> Y e. B )
6 crng32d.z 
 |-  ( ph -> Z e. B )
7 1 2 3 5 6 crngcomd 
 |-  ( ph -> ( Y .x. Z ) = ( Z .x. Y ) )
8 7 oveq2d 
 |-  ( ph -> ( X .x. ( Y .x. Z ) ) = ( X .x. ( Z .x. Y ) ) )
9 3 crngringd 
 |-  ( ph -> R e. Ring )
10 1 2 9 4 5 6 ringassd 
 |-  ( ph -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
11 1 2 9 4 6 5 ringassd 
 |-  ( ph -> ( ( X .x. Z ) .x. Y ) = ( X .x. ( Z .x. Y ) ) )
12 8 10 11 3eqtr4d 
 |-  ( ph -> ( ( X .x. Y ) .x. Z ) = ( ( X .x. Z ) .x. Y ) )