Theorem axmulf 7863

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 7860. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 7929. (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 2912	. . . . . . . . 9 |- E* z z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >.
2	1	mosubop 4690	. . . . . . . 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 4690	. . . . . . 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 1606	. . . . . . . . . 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 1911	. . . . . . . . . 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 1606	. . . . . . . 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 2063	. . . . . . 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 2096	. . . . 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 5970	. . . 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 7818	. . . . 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 5234	. . . 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 4825	. . . . 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 5952	. . . . 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 3187	. . . 4 |- dom x. C_ ( CC X. CC )
19		cnm 7826	. . . . . . 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 7860	. . . . . . 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 5230	. . . . . . 7 |- ( Fun x. -> Rel x. )
24	15, 23	mp1i 10	. . . . . 6 |- ( T. -> Rel x. )
25	20, 22, 24	oprssdmm 6167	. . . . 5 |- ( T. -> ( CC X. CC ) C_ dom x. )
26	25	mptru 1362	. . . 4 |- ( CC X. CC ) C_ dom x.
27	18, 26	eqssi 3171	. . 3 |- dom x. = ( CC X. CC )
28		df-fn 5216	. . 3 |- ( x. Fn ( CC X. CC ) <-> ( Fun x. /\ dom x. = ( CC X. CC ) ) )
29	15, 27, 28	mpbir2an 942	. 2 |- x. Fn ( CC X. CC )
30	21	rgen2a 2531	. 2 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
31		ffnov 5974	. 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 942	1 |- x. : ( CC X. CC ) --> CC

Syntax hints:   /\ wa 104   = wceq 1353   T. wtru 1354   E.wex 1492   E*wmo 2027   e. wcel 2148   A.wral 2455   C_ wss 3129   <.cop 3595   X. cxp 4622   dom cdm 4624   Rel wrel 4629   Fun wfun 5207   Fn wfn 5208   -->wf 5209  (class class class)co 5870   {coprab 5871   -1Rcm1r 7294   +R cplr 7295   .R cmr 7296   CCcc 7804   x. cmul 7811

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-eprel 4287  df-id 4291  df-po 4294  df-iso 4295  df-iord 4364  df-on 4366  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-recs 6301  df-irdg 6366  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-er 6530  df-ec 6532  df-qs 6536  df-ni 7298  df-pli 7299  df-mi 7300  df-lti 7301  df-plpq 7338  df-mpq 7339  df-enq 7341  df-nqqs 7342  df-plqqs 7343  df-mqqs 7344  df-1nqqs 7345  df-rq 7346  df-ltnqqs 7347  df-enq0 7418  df-nq0 7419  df-0nq0 7420  df-plq0 7421  df-mq0 7422  df-inp 7460  df-i1p 7461  df-iplp 7462  df-imp 7463  df-enr 7720  df-nr 7721  df-plr 7722  df-mr 7723  df-m1r 7727  df-c 7812  df-mul 7818

This theorem is referenced by: (None)