Theorem axmulf 9967

Description: Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 9974. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 10016. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
axmulf	|- x. : ( CC X. CC ) --> CC

Proof of Theorem axmulf

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

Step	Hyp	Ref	Expression
1		moeq 3382	. . . . . . . . 9 |- E* z z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >.
2	1	mosubop 4973	. . . . . . . 8 |- E* z E. u E. f ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. )
3	2	mosubop 4973	. . . . . . 7 |- E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) )
4		anass 681	. . . . . . . . . . 11 |- ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) <-> ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) )
5	4	2exbii 1775	. . . . . . . . . 10 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) <-> E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) )
6		19.42vv 1920	. . . . . . . . . 10 |- ( E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) )
7	5, 6	bitri 264	. . . . . . . . 9 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) )
8	7	2exbii 1775	. . . . . . . 8 |- ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) <-> E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) )
9	8	mobii 2493	. . . . . . 7 |- ( E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) <-> E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) )
10	3, 9	mpbir 221	. . . . . 6 |- E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. )
11	10	moani 2525	. . . . 5 |- E* z ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) )
12	11	funoprab 6760	. . . 4 |- Fun { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
13		df-mul 9948	. . . . 5 |- x. = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
14	13	funeqi 5909	. . . 4 |- ( Fun x. <-> Fun { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) } )
15	12, 14	mpbir 221	. . 3 |- Fun x.
16	13	dmeqi 5325	. . . . 5 |- dom x. = dom { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
17		dmoprabss 6742	. . . . 5 |- dom { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) } C_ ( CC X. CC )
18	16, 17	eqsstri 3635	. . . 4 |- dom x. C_ ( CC X. CC )
19		0ncn 9954	. . . . 5 |- -. (/) e. CC
20		df-c 9942	. . . . . . 7 |- CC = ( R. X. R. )
21		oveq1 6657	. . . . . . . 8 |- ( <. z , w >. = x -> ( <. z , w >. x. <. v , u >. ) = ( x x. <. v , u >. ) )
22	21	eleq1d 2686	. . . . . . 7 |- ( <. z , w >. = x -> ( ( <. z , w >. x. <. v , u >. ) e. ( R. X. R. ) <-> ( x x. <. v , u >. ) e. ( R. X. R. ) ) )
23		oveq2 6658	. . . . . . . 8 |- ( <. v , u >. = y -> ( x x. <. v , u >. ) = ( x x. y ) )
24	23	eleq1d 2686	. . . . . . 7 |- ( <. v , u >. = y -> ( ( x x. <. v , u >. ) e. ( R. X. R. ) <-> ( x x. y ) e. ( R. X. R. ) ) )
25		mulcnsr 9957	. . . . . . . 8 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( <. z , w >. x. <. v , u >. ) = <. ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) , ( ( w .R v ) +R ( z .R u ) ) >. )
26		mulclsr 9905	. . . . . . . . . . 11 |- ( ( z e. R. /\ v e. R. ) -> ( z .R v ) e. R. )
27		m1r 9903	. . . . . . . . . . . 12 |- -1R e. R.
28		mulclsr 9905	. . . . . . . . . . . 12 |- ( ( w e. R. /\ u e. R. ) -> ( w .R u ) e. R. )
29		mulclsr 9905	. . . . . . . . . . . 12 |- ( ( -1R e. R. /\ ( w .R u ) e. R. ) -> ( -1R .R ( w .R u ) ) e. R. )
30	27, 28, 29	sylancr 695	. . . . . . . . . . 11 |- ( ( w e. R. /\ u e. R. ) -> ( -1R .R ( w .R u ) ) e. R. )
31		addclsr 9904	. . . . . . . . . . 11 |- ( ( ( z .R v ) e. R. /\ ( -1R .R ( w .R u ) ) e. R. ) -> ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. )
32	26, 30, 31	syl2an 494	. . . . . . . . . 10 |- ( ( ( z e. R. /\ v e. R. ) /\ ( w e. R. /\ u e. R. ) ) -> ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. )
33	32	an4s 869	. . . . . . . . 9 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. )
34		mulclsr 9905	. . . . . . . . . . 11 |- ( ( w e. R. /\ v e. R. ) -> ( w .R v ) e. R. )
35		mulclsr 9905	. . . . . . . . . . 11 |- ( ( z e. R. /\ u e. R. ) -> ( z .R u ) e. R. )
36		addclsr 9904	. . . . . . . . . . 11 |- ( ( ( w .R v ) e. R. /\ ( z .R u ) e. R. ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
37	34, 35, 36	syl2anr 495	. . . . . . . . . 10 |- ( ( ( z e. R. /\ u e. R. ) /\ ( w e. R. /\ v e. R. ) ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
38	37	an42s 870	. . . . . . . . 9 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
39		opelxpi 5148	. . . . . . . . 9 |- ( ( ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) e. R. /\ ( ( w .R v ) +R ( z .R u ) ) e. R. ) -> <. ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) , ( ( w .R v ) +R ( z .R u ) ) >. e. ( R. X. R. ) )
40	33, 38, 39	syl2anc 693	. . . . . . . 8 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> <. ( ( z .R v ) +R ( -1R .R ( w .R u ) ) ) , ( ( w .R v ) +R ( z .R u ) ) >. e. ( R. X. R. ) )
41	25, 40	eqeltrd 2701	. . . . . . 7 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( <. z , w >. x. <. v , u >. ) e. ( R. X. R. ) )
42	20, 22, 24, 41	2optocl 5196	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. ( R. X. R. ) )
43	42, 20	syl6eleqr 2712	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
44	19, 43	oprssdm 6815	. . . 4 |- ( CC X. CC ) C_ dom x.
45	18, 44	eqssi 3619	. . 3 |- dom x. = ( CC X. CC )
46		df-fn 5891	. . 3 |- ( x. Fn ( CC X. CC ) <-> ( Fun x. /\ dom x. = ( CC X. CC ) ) )
47	15, 45, 46	mpbir2an 955	. 2 |- x. Fn ( CC X. CC )
48	43	rgen2a 2977	. 2 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
49		ffnov 6764	. 2 |- ( x. : ( CC X. CC ) --> CC <-> ( x. Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x x. y ) e. CC ) )
50	47, 48, 49	mpbir2an 955	1 |- x. : ( CC X. CC ) --> CC

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E*wmo 2471   A.wral 2912   <.cop 4183   X. cxp 5112   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   -->wf 5884  (class class class)co 6650   {coprab 6651   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:  axmulcl  9974