Theorem mpomulf 8169

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 8155 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8130. (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 8159	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
2	1	rgen2 2618	. . 3 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
3		eqid 2231	. . . 4 |- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( x e. CC , y e. CC |-> ( x x. y ) )
4	3	fnmpo 6367	. . 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 529	. . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> y e. CC )
8		eleq1a 2303	. . . . . . 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 1203	. . . 4 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> ( x e. CC /\ y e. CC /\ z e. CC ) )
12	11	ssoprab2i 6110	. . 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 6023	. . 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 6359	. . 3 |- ( ( CC X. CC ) X. CC ) = { <. <. x , y >. , z >. | ( x e. CC /\ y e. CC /\ z e. CC ) }
15	12, 13, 14	3sstr4i 3268	. 2 |- ( x e. CC , y e. CC |-> ( x x. y ) ) C_ ( ( CC X. CC ) X. CC )
16		dff2 5791	. 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 950	1 |- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   X. cxp 4723   Fn wfn 5321   -->wf 5322  (class class class)co 6018   {coprab 6019   e. cmpo 6020   CCcc 8030   x. cmul 8037

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-mulcl 8130

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304

This theorem is referenced by:  mpomulcn  15296  mpodvdsmulf1o  15720  fsumdvdsmul  15721