Theorem axmulf 8226

Description: Multiplication is an operation on the complex numbers. This is the construction-dependent version of ax-mulf 8292 and it should not be referenced outside the construction. We generally prefer to develop our theory using the less specific mulcl 8296. (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 3001	. . . . . . . . 9 |- E* z z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >.
2	1	mosubop 4836	. . . . . . . 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 4836	. . . . . . 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 405	. . . . . . . . . . 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 1659	. . . . . . . . . 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 1967	. . . . . . . . . 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 1659	. . . . . . . 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 2123	. . . . . . 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 2157	. . . . 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 6178	. . . 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 8181	. . . . 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 5393	. . . 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 4977	. . . . 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 6160	. . . . 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 3280	. . . 4 |- dom x. C_ ( CC X. CC )
19		cnm 8189	. . . . . . 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 8223	. . . . . . 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 5389	. . . . . . 7 |- ( Fun x. -> Rel x. )
24	15, 23	mp1i 10	. . . . . 6 |- ( T. -> Rel x. )
25	20, 22, 24	oprssdmm 6395	. . . . 5 |- ( T. -> ( CC X. CC ) C_ dom x. )
26	25	mptru 1411	. . . 4 |- ( CC X. CC ) C_ dom x.
27	18, 26	eqssi 3264	. . 3 |- dom x. = ( CC X. CC )
28		df-fn 5375	. . 3 |- ( x. Fn ( CC X. CC ) <-> ( Fun x. /\ dom x. = ( CC X. CC ) ) )
29	15, 27, 28	mpbir2an 955	. 2 |- x. Fn ( CC X. CC )
30	21	rgen2a 2604	. 2 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
31		ffnov 6182	. 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 955	1 |- x. : ( CC X. CC ) --> CC

Syntax hints:   /\ wa 104   = wceq 1402   T. wtru 1403   E.wex 1545   E*wmo 2087   e. wcel 2209   A.wral 2528   C_ wss 3220   <.cop 3708   X. cxp 4767   dom cdm 4769   Rel wrel 4774   Fun wfun 5366   Fn wfn 5367   -->wf 5368  (class class class)co 6075   {coprab 6076   -1Rcm1r 7657   +R cplr 7658   .R cmr 7659   CCcc 8167   x. cmul 8174

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-iplp 7825  df-imp 7826  df-enr 8083  df-nr 8084  df-plr 8085  df-mr 8086  df-m1r 8090  df-c 8175  df-mul 8181

This theorem is referenced by: (None)