Theorem axmulass 9978

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 10002. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Assertion

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

Proof of Theorem axmulass

Dummy variables x y z w v u f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcnqs 9963	. 2 |- CC = ( ( R. X. R. ) /. `' _E )
2		mulcnsrec 9965	. 2 |- ( ( ( x e. R. /\ y e. R. ) /\ ( z e. R. /\ w e. R. ) ) -> ( [ <. x , y >. ] `' _E x. [ <. z , w >. ] `' _E ) = [ <. ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) , ( ( y .R z ) +R ( x .R w ) ) >. ] `' _E )
3		mulcnsrec 9965	. 2 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( [ <. z , w >. ] `' _E x. [ <. v , u >. ] `' _E ) = [ <. ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) , ( ( w .R v ) +R ( z .R u ) ) >. ] `' _E )
4		mulcnsrec 9965	. 2 |- ( ( ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) e. R. /\ ( ( y .R z ) +R ( x .R w ) ) e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( [ <. ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) , ( ( y .R z ) +R ( x .R w ) ) >. ] `' _E x. [ <. v , u >. ] `' _E ) = [ <. ( ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R v ) +R ( -1R .R ( ( ( y .R z ) +R ( x .R w ) ) .R u ) ) ) , ( ( ( ( y .R z ) +R ( x .R w ) ) .R v ) +R ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R u ) ) >. ] `' _E )
5		mulcnsrec 9965	. 2 |- ( ( ( x e. R. /\ y e. R. ) /\ ( ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. /\ ( ( w .R v ) +R ( z .R u ) ) e. R. ) ) -> ( [ <. x , y >. ] `' _E x. [ <. ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) , ( ( w .R v ) +R ( z .R u ) ) >. ] `' _E ) = [ <. ( ( x .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) +R ( -1R .R ( y .R ( ( w .R v ) +R ( z .R u ) ) ) ) ) , ( ( y .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) +R ( x .R ( ( w .R v ) +R ( z .R u ) ) ) ) >. ] `' _E )
6		mulclsr 9905	. . . . 5 |- ( ( x e. R. /\ z e. R. ) -> ( x .R z ) e. R. )
7		m1r 9903	. . . . . 6 |- -1R e. R.
8		mulclsr 9905	. . . . . 6 |- ( ( y e. R. /\ w e. R. ) -> ( y .R w ) e. R. )
9		mulclsr 9905	. . . . . 6 |- ( ( -1R e. R. /\ ( y .R w ) e. R. ) -> ( -1R .R ( y .R w ) ) e. R. )
10	7, 8, 9	sylancr 695	. . . . 5 |- ( ( y e. R. /\ w e. R. ) -> ( -1R .R ( y .R w ) ) e. R. )
11		addclsr 9904	. . . . 5 |- ( ( ( x .R z ) e. R. /\ ( -1R .R ( y .R w ) ) e. R. ) -> ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) e. R. )
12	6, 10, 11	syl2an 494	. . . 4 |- ( ( ( x e. R. /\ z e. R. ) /\ ( y e. R. /\ w e. R. ) ) -> ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) e. R. )
13	12	an4s 869	. . 3 |- ( ( ( x e. R. /\ y e. R. ) /\ ( z e. R. /\ w e. R. ) ) -> ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) e. R. )
14		mulclsr 9905	. . . . 5 |- ( ( y e. R. /\ z e. R. ) -> ( y .R z ) e. R. )
15		mulclsr 9905	. . . . 5 |- ( ( x e. R. /\ w e. R. ) -> ( x .R w ) e. R. )
16		addclsr 9904	. . . . 5 |- ( ( ( y .R z ) e. R. /\ ( x .R w ) e. R. ) -> ( ( y .R z ) +R ( x .R w ) ) e. R. )
17	14, 15, 16	syl2anr 495	. . . 4 |- ( ( ( x e. R. /\ w e. R. ) /\ ( y e. R. /\ z e. R. ) ) -> ( ( y .R z ) +R ( x .R w ) ) e. R. )
18	17	an42s 870	. . 3 |- ( ( ( x e. R. /\ y e. R. ) /\ ( z e. R. /\ w e. R. ) ) -> ( ( y .R z ) +R ( x .R w ) ) e. R. )
19	13, 18	jca 554	. 2 |- ( ( ( x e. R. /\ y e. R. ) /\ ( z e. R. /\ w e. R. ) ) -> ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) e. R. /\ ( ( y .R z ) +R ( x .R w ) ) e. R. ) )
20		mulclsr 9905	. . . . 5 |- ( ( z e. R. /\ v e. R. ) -> ( z .R v ) e. R. )
21		mulclsr 9905	. . . . . 6 |- ( ( w e. R. /\ u e. R. ) -> ( w .R u ) e. R. )
22		mulclsr 9905	. . . . . 6 |- ( ( -1R e. R. /\ ( w .R u ) e. R. ) -> ( -1R .R ( w .R u ) ) e. R. )
23	7, 21, 22	sylancr 695	. . . . 5 |- ( ( w e. R. /\ u e. R. ) -> ( -1R .R ( w .R u ) ) e. R. )
24		addclsr 9904	. . . . 5 |- ( ( ( z .R v ) e. R. /\ ( -1R .R ( w .R u ) ) e. R. ) -> ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. )
25	20, 23, 24	syl2an 494	. . . 4 |- ( ( ( z e. R. /\ v e. R. ) /\ ( w e. R. /\ u e. R. ) ) -> ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. )
26	25	an4s 869	. . 3 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. )
27		mulclsr 9905	. . . . 5 |- ( ( w e. R. /\ v e. R. ) -> ( w .R v ) e. R. )
28		mulclsr 9905	. . . . 5 |- ( ( z e. R. /\ u e. R. ) -> ( z .R u ) e. R. )
29		addclsr 9904	. . . . 5 |- ( ( ( w .R v ) e. R. /\ ( z .R u ) e. R. ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
30	27, 28, 29	syl2anr 495	. . . 4 |- ( ( ( z e. R. /\ u e. R. ) /\ ( w e. R. /\ v e. R. ) ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
31	30	an42s 870	. . 3 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
32	26, 31	jca 554	. 2 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. /\ ( ( w .R v ) +R ( z .R u ) ) e. R. ) )
33		ovex 6678	. . . 4 |- ( x .R ( z .R v ) ) e. _V
34		ovex 6678	. . . 4 |- ( x .R ( -1R .R ( w .R u ) ) ) e. _V
35		ovex 6678	. . . 4 |- ( -1R .R ( y .R ( w .R v ) ) ) e. _V
36		addcomsr 9908	. . . 4 |- ( f +R g ) = ( g +R f )
37		addasssr 9909	. . . 4 |- ( ( f +R g ) +R h ) = ( f +R ( g +R h ) )
38		ovex 6678	. . . 4 |- ( -1R .R ( y .R ( z .R u ) ) ) e. _V
39	33, 34, 35, 36, 37, 38	caov42 6867	. . 3 |- ( ( ( x .R ( z .R v ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) ) +R ( ( -1R .R ( y .R ( w .R v ) ) ) +R ( -1R .R ( y .R ( z .R u ) ) ) ) ) = ( ( ( x .R ( z .R v ) ) +R ( -1R .R ( y .R ( w .R v ) ) ) ) +R ( ( -1R .R ( y .R ( z .R u ) ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) ) )
40		distrsr 9912	. . . 4 |- ( x .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) = ( ( x .R ( z .R v ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) )
41		distrsr 9912	. . . . . 6 |- ( y .R ( ( w .R v ) +R ( z .R u ) ) ) = ( ( y .R ( w .R v ) ) +R ( y .R ( z .R u ) ) )
42	41	oveq2i 6661	. . . . 5 |- ( -1R .R ( y .R ( ( w .R v ) +R ( z .R u ) ) ) ) = ( -1R .R ( ( y .R ( w .R v ) ) +R ( y .R ( z .R u ) ) ) )
43		distrsr 9912	. . . . 5 |- ( -1R .R ( ( y .R ( w .R v ) ) +R ( y .R ( z .R u ) ) ) ) = ( ( -1R .R ( y .R ( w .R v ) ) ) +R ( -1R .R ( y .R ( z .R u ) ) ) )
44	42, 43	eqtri 2644	. . . 4 |- ( -1R .R ( y .R ( ( w .R v ) +R ( z .R u ) ) ) ) = ( ( -1R .R ( y .R ( w .R v ) ) ) +R ( -1R .R ( y .R ( z .R u ) ) ) )
45	40, 44	oveq12i 6662	. . 3 |- ( ( x .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) +R ( -1R .R ( y .R ( ( w .R v ) +R ( z .R u ) ) ) ) ) = ( ( ( x .R ( z .R v ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) ) +R ( ( -1R .R ( y .R ( w .R v ) ) ) +R ( -1R .R ( y .R ( z .R u ) ) ) ) )
46		vex 3203	. . . . . 6 |- x e. _V
47	7	elexi 3213	. . . . . 6 |- -1R e. _V
48		vex 3203	. . . . . 6 |- z e. _V
49		mulcomsr 9910	. . . . . 6 |- ( f .R g ) = ( g .R f )
50		distrsr 9912	. . . . . 6 |- ( f .R ( g +R h ) ) = ( ( f .R g ) +R ( f .R h ) )
51		ovex 6678	. . . . . 6 |- ( y .R w ) e. _V
52		vex 3203	. . . . . 6 |- v e. _V
53		mulasssr 9911	. . . . . 6 |- ( ( f .R g ) .R h ) = ( f .R ( g .R h ) )
54	46, 47, 48, 49, 50, 51, 52, 53	caovdilem 6869	. . . . 5 |- ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R v ) = ( ( x .R ( z .R v ) ) +R ( -1R .R ( ( y .R w ) .R v ) ) )
55		mulasssr 9911	. . . . . . 7 |- ( ( y .R w ) .R v ) = ( y .R ( w .R v ) )
56	55	oveq2i 6661	. . . . . 6 |- ( -1R .R ( ( y .R w ) .R v ) ) = ( -1R .R ( y .R ( w .R v ) ) )
57	56	oveq2i 6661	. . . . 5 |- ( ( x .R ( z .R v ) ) +R ( -1R .R ( ( y .R w ) .R v ) ) ) = ( ( x .R ( z .R v ) ) +R ( -1R .R ( y .R ( w .R v ) ) ) )
58	54, 57	eqtri 2644	. . . 4 |- ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R v ) = ( ( x .R ( z .R v ) ) +R ( -1R .R ( y .R ( w .R v ) ) ) )
59		vex 3203	. . . . . . 7 |- y e. _V
60		vex 3203	. . . . . . 7 |- w e. _V
61		vex 3203	. . . . . . 7 |- u e. _V
62	59, 46, 48, 49, 50, 60, 61, 53	caovdilem 6869	. . . . . 6 |- ( ( ( y .R z ) +R ( x .R w ) ) .R u ) = ( ( y .R ( z .R u ) ) +R ( x .R ( w .R u ) ) )
63	62	oveq2i 6661	. . . . 5 |- ( -1R .R ( ( ( y .R z ) +R ( x .R w ) ) .R u ) ) = ( -1R .R ( ( y .R ( z .R u ) ) +R ( x .R ( w .R u ) ) ) )
64		distrsr 9912	. . . . . 6 |- ( -1R .R ( ( y .R ( z .R u ) ) +R ( x .R ( w .R u ) ) ) ) = ( ( -1R .R ( y .R ( z .R u ) ) ) +R ( -1R .R ( x .R ( w .R u ) ) ) )
65		ovex 6678	. . . . . . . 8 |- ( w .R u ) e. _V
66	47, 46, 65, 49, 53	caov12 6862	. . . . . . 7 |- ( -1R .R ( x .R ( w .R u ) ) ) = ( x .R ( -1R .R ( w .R u ) ) )
67	66	oveq2i 6661	. . . . . 6 |- ( ( -1R .R ( y .R ( z .R u ) ) ) +R ( -1R .R ( x .R ( w .R u ) ) ) ) = ( ( -1R .R ( y .R ( z .R u ) ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) )
68	64, 67	eqtri 2644	. . . . 5 |- ( -1R .R ( ( y .R ( z .R u ) ) +R ( x .R ( w .R u ) ) ) ) = ( ( -1R .R ( y .R ( z .R u ) ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) )
69	63, 68	eqtri 2644	. . . 4 |- ( -1R .R ( ( ( y .R z ) +R ( x .R w ) ) .R u ) ) = ( ( -1R .R ( y .R ( z .R u ) ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) )
70	58, 69	oveq12i 6662	. . 3 |- ( ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R v ) +R ( -1R .R ( ( ( y .R z ) +R ( x .R w ) ) .R u ) ) ) = ( ( ( x .R ( z .R v ) ) +R ( -1R .R ( y .R ( w .R v ) ) ) ) +R ( ( -1R .R ( y .R ( z .R u ) ) ) +R ( x .R ( -1R .R ( w .R u ) ) ) ) )
71	39, 45, 70	3eqtr4ri 2655	. 2 |- ( ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R v ) +R ( -1R .R ( ( ( y .R z ) +R ( x .R w ) ) .R u ) ) ) = ( ( x .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) +R ( -1R .R ( y .R ( ( w .R v ) +R ( z .R u ) ) ) ) )
72		ovex 6678	. . . 4 |- ( y .R ( z .R v ) ) e. _V
73		ovex 6678	. . . 4 |- ( y .R ( -1R .R ( w .R u ) ) ) e. _V
74		ovex 6678	. . . 4 |- ( x .R ( w .R v ) ) e. _V
75		ovex 6678	. . . 4 |- ( x .R ( z .R u ) ) e. _V
76	72, 73, 74, 36, 37, 75	caov42 6867	. . 3 |- ( ( ( y .R ( z .R v ) ) +R ( y .R ( -1R .R ( w .R u ) ) ) ) +R ( ( x .R ( w .R v ) ) +R ( x .R ( z .R u ) ) ) ) = ( ( ( y .R ( z .R v ) ) +R ( x .R ( w .R v ) ) ) +R ( ( x .R ( z .R u ) ) +R ( y .R ( -1R .R ( w .R u ) ) ) ) )
77		distrsr 9912	. . . 4 |- ( y .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) = ( ( y .R ( z .R v ) ) +R ( y .R ( -1R .R ( w .R u ) ) ) )
78		distrsr 9912	. . . 4 |- ( x .R ( ( w .R v ) +R ( z .R u ) ) ) = ( ( x .R ( w .R v ) ) +R ( x .R ( z .R u ) ) )
79	77, 78	oveq12i 6662	. . 3 |- ( ( y .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) +R ( x .R ( ( w .R v ) +R ( z .R u ) ) ) ) = ( ( ( y .R ( z .R v ) ) +R ( y .R ( -1R .R ( w .R u ) ) ) ) +R ( ( x .R ( w .R v ) ) +R ( x .R ( z .R u ) ) ) )
80	59, 46, 48, 49, 50, 60, 52, 53	caovdilem 6869	. . . 4 |- ( ( ( y .R z ) +R ( x .R w ) ) .R v ) = ( ( y .R ( z .R v ) ) +R ( x .R ( w .R v ) ) )
81	46, 47, 48, 49, 50, 51, 61, 53	caovdilem 6869	. . . . 5 |- ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R u ) = ( ( x .R ( z .R u ) ) +R ( -1R .R ( ( y .R w ) .R u ) ) )
82		mulasssr 9911	. . . . . . . 8 |- ( ( y .R w ) .R u ) = ( y .R ( w .R u ) )
83	82	oveq2i 6661	. . . . . . 7 |- ( -1R .R ( ( y .R w ) .R u ) ) = ( -1R .R ( y .R ( w .R u ) ) )
84	47, 59, 65, 49, 53	caov12 6862	. . . . . . 7 |- ( -1R .R ( y .R ( w .R u ) ) ) = ( y .R ( -1R .R ( w .R u ) ) )
85	83, 84	eqtri 2644	. . . . . 6 |- ( -1R .R ( ( y .R w ) .R u ) ) = ( y .R ( -1R .R ( w .R u ) ) )
86	85	oveq2i 6661	. . . . 5 |- ( ( x .R ( z .R u ) ) +R ( -1R .R ( ( y .R w ) .R u ) ) ) = ( ( x .R ( z .R u ) ) +R ( y .R ( -1R .R ( w .R u ) ) ) )
87	81, 86	eqtri 2644	. . . 4 |- ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R u ) = ( ( x .R ( z .R u ) ) +R ( y .R ( -1R .R ( w .R u ) ) ) )
88	80, 87	oveq12i 6662	. . 3 |- ( ( ( ( y .R z ) +R ( x .R w ) ) .R v ) +R ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R u ) ) = ( ( ( y .R ( z .R v ) ) +R ( x .R ( w .R v ) ) ) +R ( ( x .R ( z .R u ) ) +R ( y .R ( -1R .R ( w .R u ) ) ) ) )
89	76, 79, 88	3eqtr4ri 2655	. 2 |- ( ( ( ( y .R z ) +R ( x .R w ) ) .R v ) +R ( ( ( x .R z ) +R ( -1R .R ( y .R w ) ) ) .R u ) ) = ( ( y .R ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) ) +R ( x .R ( ( w .R v ) +R ( z .R u ) ) ) )
90	1, 2, 3, 4, 5, 19, 32, 71, 89	ecovass 7855	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _E cep 5028   `'ccnv 5113  (class class class)co 6650   R.cnr 9687   -1Rcm1r 9690   +R cplr 9691   .R cmr 9692   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-mr 9880  df-m1r 9884  df-c 9942  df-mul 9948

This theorem is referenced by: (None)