Theorem int-mulcomd 38479

Description: MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)

Hypotheses

Ref	Expression
int-mulcomd.1	|- ( ph -> B e. RR )
int-mulcomd.2	|- ( ph -> C e. RR )
int-mulcomd.3	|- ( ph -> A = B )

Assertion

Ref	Expression
int-mulcomd	|- ( ph -> ( B x. C ) = ( C x. A ) )

Proof of Theorem int-mulcomd

Step	Hyp	Ref	Expression
1		int-mulcomd.1	. . . 4 |- ( ph -> B e. RR )
2	1	recnd 10068	. . 3 |- ( ph -> B e. CC )
3		int-mulcomd.2	. . . 4 |- ( ph -> C e. RR )
4	3	recnd 10068	. . 3 |- ( ph -> C e. CC )
5	2, 4	mulcomd 10061	. 2 |- ( ph -> ( B x. C ) = ( C x. B ) )
6		int-mulcomd.3	. . . 4 |- ( ph -> A = B )
7	6	eqcomd 2628	. . 3 |- ( ph -> B = A )
8	7	oveq2d 6666	. 2 |- ( ph -> ( C x. B ) = ( C x. A ) )
9	5, 8	eqtrd 2656	1 |- ( ph -> ( B x. C ) = ( C x. A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990  (class class class)co 6650   RRcr 9935   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-resscn 9993  ax-mulcom 10000

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)