Theorem mpomulf 8152

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 8138 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8113. (Contributed by GG, 16-Mar-2025.)

Assertion

Ref	Expression
mpomulf	|- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC

Distinct variable group:   x, y

Proof of Theorem mpomulf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 8142	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
2	1	rgen2 2616	. . 3 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
3		eqid 2229	. . . 4 |- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( x e. CC , y e. CC |-> ( x x. y ) )
4	3	fnmpo 6359	. . 3 |- ( A. x e. CC A. y e. CC ( x x. y ) e. CC -> ( x e. CC , y e. CC |-> ( x x. y ) ) Fn ( CC X. CC ) )
5	2, 4	ax-mp 5	. 2 |- ( x e. CC , y e. CC |-> ( x x. y ) ) Fn ( CC X. CC )
6		simpll 527	. . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> x e. CC )
7		simplr 528	. . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> y e. CC )
8		eleq1a 2301	. . . . . . 7 |- ( ( x x. y ) e. CC -> ( z = ( x x. y ) -> z e. CC ) )
9	1, 8	syl 14	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( z = ( x x. y ) -> z e. CC ) )
10	9	imp 124	. . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> z e. CC )
11	6, 7, 10	3jca 1201	. . . 4 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> ( x e. CC /\ y e. CC /\ z e. CC ) )
12	11	ssoprab2i 6102	. . 3 |- { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) } C_ { <. <. x , y >. , z >. | ( x e. CC /\ y e. CC /\ z e. CC ) }
13		df-mpo 6015	. . 3 |- ( x e. CC , y e. CC |-> ( x x. y ) ) = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) }
14		dfxp3 6351	. . 3 |- ( ( CC X. CC ) X. CC ) = { <. <. x , y >. , z >. | ( x e. CC /\ y e. CC /\ z e. CC ) }
15	12, 13, 14	3sstr4i 3265	. 2 |- ( x e. CC , y e. CC |-> ( x x. y ) ) C_ ( ( CC X. CC ) X. CC )
16		dff2 5784	. 2 |- ( ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC <-> ( ( x e. CC , y e. CC |-> ( x x. y ) ) Fn ( CC X. CC ) /\ ( x e. CC , y e. CC |-> ( x x. y ) ) C_ ( ( CC X. CC ) X. CC ) ) )
17	5, 15, 16	mpbir2an 948	1 |- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   X. cxp 4718   Fn wfn 5316   -->wf 5317  (class class class)co 6010   {coprab 6011   e. cmpo 6012   CCcc 8013   x. cmul 8020

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-mulcl 8113

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fo 5327  df-fv 5329  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296

This theorem is referenced by:  mpomulcn  15261  mpodvdsmulf1o  15685  fsumdvdsmul  15686