Theorem axmulf 7936

Description: Multiplication is an operation on the complex numbers. This is the construction-dependent version of ax-mulf 8002 and it should not be referenced outside the construction. We generally prefer to develop our theory using the less specific mulcl 8006. (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 a b x y z w v u f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		moeq 2939	. . . . . . . . 9 |- E* z z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >.
2	1	mosubop 4729	. . . . . . . 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 4729	. . . . . . 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 401	. . . . . . . . . . 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 1620	. . . . . . . . . 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 1926	. . . . . . . . . 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 184	. . . . . . . . 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 1620	. . . . . . . 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 2082	. . . . . . 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 146	. . . . . 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 2115	. . . . 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 6022	. . . 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 7891	. . . . 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 5279	. . . 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 146	. . 3 |- Fun x.
16	13	dmeqi 4867	. . . . 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 6004	. . . . 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 3215	. . . 4 |- dom x. C_ ( CC X. CC )
19		cnm 7899	. . . . . . 7 |- ( a e. CC -> E. b b e. a )
20	19	adantl 277	. . . . . 6 |- ( ( T. /\ a e. CC ) -> E. b b e. a )
21		axmulcl 7933	. . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
22	21	adantl 277	. . . . . 6 |- ( ( T. /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
23		funrel 5275	. . . . . . 7 |- ( Fun x. -> Rel x. )
24	15, 23	mp1i 10	. . . . . 6 |- ( T. -> Rel x. )
25	20, 22, 24	oprssdmm 6229	. . . . 5 |- ( T. -> ( CC X. CC ) C_ dom x. )
26	25	mptru 1373	. . . 4 |- ( CC X. CC ) C_ dom x.
27	18, 26	eqssi 3199	. . 3 |- dom x. = ( CC X. CC )
28		df-fn 5261	. . 3 |- ( x. Fn ( CC X. CC ) <-> ( Fun x. /\ dom x. = ( CC X. CC ) ) )
29	15, 27, 28	mpbir2an 944	. 2 |- x. Fn ( CC X. CC )
30	21	rgen2a 2551	. 2 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
31		ffnov 6026	. 2 |- ( x. : ( CC X. CC ) --> CC <-> ( x. Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x x. y ) e. CC ) )
32	29, 30, 31	mpbir2an 944	1 |- x. : ( CC X. CC ) --> CC

Syntax hints:   /\ wa 104   = wceq 1364   T. wtru 1365   E.wex 1506   E*wmo 2046   e. wcel 2167   A.wral 2475   C_ wss 3157   <.cop 3625   X. cxp 4661   dom cdm 4663   Rel wrel 4668   Fun wfun 5252   Fn wfn 5253   -->wf 5254  (class class class)co 5922   {coprab 5923   -1Rcm1r 7367   +R cplr 7368   .R cmr 7369   CCcc 7877   x. cmul 7884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536  df-enr 7793  df-nr 7794  df-plr 7795  df-mr 7796  df-m1r 7800  df-c 7885  df-mul 7891

This theorem is referenced by: (None)