Theorem axmulf 9514

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 9521. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 9563. (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 3274	. . . . . . . . 9 |- E* z z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >.
2	1	mosubop 4741	. . . . . . . 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 4741	. . . . . . 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 649	. . . . . . . . . . 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 1640	. . . . . . . . . 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 1946	. . . . . . . . . 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 249	. . . . . . . . 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 1640	. . . . . . . 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 2296	. . . . . . 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 209	. . . . . 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 2343	. . . . 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 6379	. . . 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 9495	. . . . 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 5601	. . . 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 209	. . 3 |- Fun x.
16	13	dmeqi 5197	. . . . 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 6361	. . . . 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 3529	. . . 4 |- dom x. C_ ( CC X. CC )
19		0ncn 9501	. . . . 5 |- -. (/) e. CC
20		df-c 9489	. . . . . . 7 |- CC = ( R. X. R. )
21		oveq1 6284	. . . . . . . 8 |- ( <. z , w >. = x -> ( <. z , w >. x. <. v , u >. ) = ( x x. <. v , u >. ) )
22	21	eleq1d 2531	. . . . . . 7 |- ( <. z , w >. = x -> ( ( <. z , w >. x. <. v , u >. ) e. ( R. X. R. ) <-> ( x x. <. v , u >. ) e. ( R. X. R. ) ) )
23		oveq2 6285	. . . . . . . 8 |- ( <. v , u >. = y -> ( x x. <. v , u >. ) = ( x x. y ) )
24	23	eleq1d 2531	. . . . . . 7 |- ( <. v , u >. = y -> ( ( x x. <. v , u >. ) e. ( R. X. R. ) <-> ( x x. y ) e. ( R. X. R. ) ) )
25		mulcnsr 9504	. . . . . . . 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 9452	. . . . . . . . . . 11 |- ( ( z e. R. /\ v e. R. ) -> ( z .R v ) e. R. )
27		m1r 9450	. . . . . . . . . . . 12 |- -1R e. R.
28		mulclsr 9452	. . . . . . . . . . . 12 |- ( ( w e. R. /\ u e. R. ) -> ( w .R u ) e. R. )
29		mulclsr 9452	. . . . . . . . . . . 12 |- ( ( -1R e. R. /\ ( w .R u ) e. R. ) -> ( -1R .R ( w .R u ) ) e. R. )
30	27, 28, 29	sylancr 663	. . . . . . . . . . 11 |- ( ( w e. R. /\ u e. R. ) -> ( -1R .R ( w .R u ) ) e. R. )
31		addclsr 9451	. . . . . . . . . . 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 477	. . . . . . . . . 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 823	. . . . . . . . 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 9452	. . . . . . . . . . 11 |- ( ( w e. R. /\ v e. R. ) -> ( w .R v ) e. R. )
35		mulclsr 9452	. . . . . . . . . . 11 |- ( ( z e. R. /\ u e. R. ) -> ( z .R u ) e. R. )
36		addclsr 9451	. . . . . . . . . . 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 478	. . . . . . . . . 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 824	. . . . . . . . 9 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
39		opelxpi 5025	. . . . . . . . 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 661	. . . . . . . 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 2550	. . . . . . 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 5070	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. ( R. X. R. ) )
43	42, 20	syl6eleqr 2561	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
44	19, 43	oprssdm 6433	. . . 4 |- ( CC X. CC ) C_ dom x.
45	18, 44	eqssi 3515	. . 3 |- dom x. = ( CC X. CC )
46		df-fn 5584	. . 3 |- ( x. Fn ( CC X. CC ) <-> ( Fun x. /\ dom x. = ( CC X. CC ) ) )
47	15, 45, 46	mpbir2an 913	. 2 |- x. Fn ( CC X. CC )
48	43	rgen2a 2886	. 2 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
49		ffnov 6383	. 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 913	1 |- x. : ( CC X. CC ) --> CC

Syntax hints:   /\ wa 369   = wceq 1374   E.wex 1591   e. wcel 1762   E*wmo 2271   A.wral 2809   <.cop 4028   X. cxp 4992   dom cdm 4994   Fun wfun 5575   Fn wfn 5576   -->wf 5577  (class class class)co 6277   {coprab 6278   R.cnr 9234   -1Rcm1r 9237   +R cplr 9238   .R cmr 9239   CCcc 9481   x. cmul 9488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-ec 7305  df-qs 7309  df-ni 9241  df-pli 9242  df-mi 9243  df-lti 9244  df-plpq 9277  df-mpq 9278  df-ltpq 9279  df-enq 9280  df-nq 9281  df-erq 9282  df-plq 9283  df-mq 9284  df-1nq 9285  df-rq 9286  df-ltnq 9287  df-np 9350  df-1p 9351  df-plp 9352  df-mp 9353  df-ltp 9354  df-enr 9424  df-nr 9425  df-plr 9426  df-mr 9427  df-m1r 9431  df-c 9489  df-mul 9495

This theorem is referenced by:  axmulcl  9521