Theorem mul4 10205

Description: Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

Assertion

Ref	Expression
mul4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )

Proof of Theorem mul4

Step	Hyp	Ref	Expression
1		mul32 10203	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )
2	1	oveq1d 6665	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A x. B ) x. C ) x. D ) = ( ( ( A x. C ) x. B ) x. D ) )
3	2	3expa 1265	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( ( A x. B ) x. C ) x. D ) = ( ( ( A x. C ) x. B ) x. D ) )
4	3	adantrr 753	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A x. B ) x. C ) x. D ) = ( ( ( A x. C ) x. B ) x. D ) )
5		mulcl 10020	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
6		mulass 10024	. . . 4 |- ( ( ( A x. B ) e. CC /\ C e. CC /\ D e. CC ) -> ( ( ( A x. B ) x. C ) x. D ) = ( ( A x. B ) x. ( C x. D ) ) )
7	6	3expb 1266	. . 3 |- ( ( ( A x. B ) e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A x. B ) x. C ) x. D ) = ( ( A x. B ) x. ( C x. D ) ) )
8	5, 7	sylan 488	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A x. B ) x. C ) x. D ) = ( ( A x. B ) x. ( C x. D ) ) )
9		mulcl 10020	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
10		mulass 10024	. . . . 5 |- ( ( ( A x. C ) e. CC /\ B e. CC /\ D e. CC ) -> ( ( ( A x. C ) x. B ) x. D ) = ( ( A x. C ) x. ( B x. D ) ) )
11	10	3expb 1266	. . . 4 |- ( ( ( A x. C ) e. CC /\ ( B e. CC /\ D e. CC ) ) -> ( ( ( A x. C ) x. B ) x. D ) = ( ( A x. C ) x. ( B x. D ) ) )
12	9, 11	sylan 488	. . 3 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( ( A x. C ) x. B ) x. D ) = ( ( A x. C ) x. ( B x. D ) ) )
13	12	an4s 869	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A x. C ) x. B ) x. D ) = ( ( A x. C ) x. ( B x. D ) ) )
14	4, 8, 13	3eqtr3d 2664	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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:  mul4i  10233  mul4d  10248  recextlem1  10657  divmuldiv  10725  mulexp  12899  demoivreALT  14931  bposlem9  25017