Theorem mul13d 39491

Description: Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
mul13d.1	|- ( ph -> A e. CC )
mul13d.2	|- ( ph -> B e. CC )
mul13d.3	|- ( ph -> C e. CC )

Assertion

Ref	Expression
mul13d	|- ( ph -> ( A x. ( B x. C ) ) = ( C x. ( B x. A ) ) )

Proof of Theorem mul13d

Step	Hyp	Ref	Expression
1		mul13d.1	. . 3 |- ( ph -> A e. CC )
2		mul13d.2	. . 3 |- ( ph -> B e. CC )
3		mul13d.3	. . 3 |- ( ph -> C e. CC )
4	1, 2, 3	mul12d 10245	. 2 |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
5	2, 1, 3	mulassd 10063	. 2 |- ( ph -> ( ( B x. A ) x. C ) = ( B x. ( A x. C ) ) )
6	2, 1	mulcld 10060	. . 3 |- ( ph -> ( B x. A ) e. CC )
7	6, 3	mulcomd 10061	. 2 |- ( ph -> ( ( B x. A ) x. C ) = ( C x. ( B x. A ) ) )
8	4, 5, 7	3eqtr2d 2662	1 |- ( ph -> ( A x. ( B x. C ) ) = ( C x. ( B x. A ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   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-mulcl 9998  ax-mulcom 10000  ax-mulass 10002

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:  dirkertrigeqlem3  40317  fourierdlem83  40406