Theorem mpomulf 8316

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 8302 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8277. (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 8306	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
2	1	rgen2 2636	. . 3 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
3		eqid 2238	. . . 4 |- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( x e. CC , y e. CC |-> ( x x. y ) )
4	3	fnmpo 6438	. . 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 531	. . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> x e. CC )
7		simplr 533	. . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> y e. CC )
8		eleq1a 2310	. . . . . . 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 1208	. . . 4 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> ( x e. CC /\ y e. CC /\ z e. CC ) )
12	11	ssoprab2i 6177	. . 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 6090	. . 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 6430	. . 3 |- ( ( CC X. CC ) X. CC ) = { <. <. x , y >. , z >. | ( x e. CC /\ y e. CC /\ z e. CC ) }
15	12, 13, 14	3sstr4i 3289	. 2 |- ( x e. CC , y e. CC |-> ( x x. y ) ) C_ ( ( CC X. CC ) X. CC )
16		dff2 5852	. 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 955	1 |- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   X. cxp 4772   Fn wfn 5372   -->wf 5373  (class class class)co 6085   {coprab 6086   e. cmpo 6087   CCcc 8177   x. cmul 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-mulcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375

This theorem is used by:  mpomulcn  15669  mpodvdsmulf1o  16110  fsumdvdsmul  16111