Theorem axmulf 9521

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 9528. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 9570. (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 3259	. . . . . . . . 9 |- E* z z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >.
2	1	mosubop 4732	. . . . . . . 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 4732	. . . . . . 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 1653	. . . . . . . . . 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 1761	. . . . . . . . . 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 1653	. . . . . . . 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 2291	. . . . . . 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 2330	. . . . 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 6383	. . . 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 9502	. . . . 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 5594	. . . 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 5190	. . . . 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 6365	. . . . 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 3516	. . . 4 |- dom x. C_ ( CC X. CC )
19		0ncn 9508	. . . . 5 |- -. (/) e. CC
20		df-c 9496	. . . . . . 7 |- CC = ( R. X. R. )
21		oveq1 6284	. . . . . . . 8 |- ( <. z , w >. = x -> ( <. z , w >. x. <. v , u >. ) = ( x x. <. v , u >. ) )
22	21	eleq1d 2510	. . . . . . 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 2510	. . . . . . 7 |- ( <. v , u >. = y -> ( ( x x. <. v , u >. ) e. ( R. X. R. ) <-> ( x x. y ) e. ( R. X. R. ) ) )
25		mulcnsr 9511	. . . . . . . 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 9459	. . . . . . . . . . 11 |- ( ( z e. R. /\ v e. R. ) -> ( z .R v ) e. R. )
27		m1r 9457	. . . . . . . . . . . 12 |- -1R e. R.
28		mulclsr 9459	. . . . . . . . . . . 12 |- ( ( w e. R. /\ u e. R. ) -> ( w .R u ) e. R. )
29		mulclsr 9459	. . . . . . . . . . . 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 9458	. . . . . . . . . . 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 824	. . . . . . . . 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 9459	. . . . . . . . . . 11 |- ( ( w e. R. /\ v e. R. ) -> ( w .R v ) e. R. )
35		mulclsr 9459	. . . . . . . . . . 11 |- ( ( z e. R. /\ u e. R. ) -> ( z .R u ) e. R. )
36		addclsr 9458	. . . . . . . . . . 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 825	. . . . . . . . 9 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( w .R v ) +R ( z .R u ) ) e. R. )
39		opelxpi 5017	. . . . . . . . 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 2529	. . . . . . 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 5063	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. ( R. X. R. ) )
43	42, 20	syl6eleqr 2540	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
44	19, 43	oprssdm 6437	. . . 4 |- ( CC X. CC ) C_ dom x.
45	18, 44	eqssi 3502	. . 3 |- dom x. = ( CC X. CC )
46		df-fn 5577	. . 3 |- ( x. Fn ( CC X. CC ) <-> ( Fun x. /\ dom x. = ( CC X. CC ) ) )
47	15, 45, 46	mpbir2an 918	. 2 |- x. Fn ( CC X. CC )
48	43	rgen2a 2868	. 2 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
49		ffnov 6387	. 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 918	1 |- x. : ( CC X. CC ) --> CC

Syntax hints:   /\ wa 369   = wceq 1381   E.wex 1597   e. wcel 1802   E*wmo 2267   A.wral 2791   <.cop 4016   X. cxp 4983   dom cdm 4985   Fun wfun 5568   Fn wfn 5569   -->wf 5570  (class class class)co 6277   {coprab 6278   R.cnr 9241   -1Rcm1r 9244   +R cplr 9245   .R cmr 9246   CCcc 9488   x. cmul 9495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-omul 7133  df-er 7309  df-ec 7311  df-qs 7315  df-ni 9248  df-pli 9249  df-mi 9250  df-lti 9251  df-plpq 9284  df-mpq 9285  df-ltpq 9286  df-enq 9287  df-nq 9288  df-erq 9289  df-plq 9290  df-mq 9291  df-1nq 9292  df-rq 9293  df-ltnq 9294  df-np 9357  df-1p 9358  df-plp 9359  df-mp 9360  df-ltp 9361  df-enr 9431  df-nr 9432  df-plr 9433  df-mr 9434  df-m1r 9438  df-c 9496  df-mul 9502

This theorem is referenced by:  axmulcl  9528