Theorem mpomulf 8009

Description: Multiplication is an operation on complex numbers. Version of ax-mulf 7995 using maps-to notation, proved from the axioms of set theory and ax-mulcl 7970. (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 7999	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
2	1	rgen2 2580	. . 3 |- A. x e. CC A. y e. CC ( x x. y ) e. CC
3		eqid 2193	. . . 4 |- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( x e. CC , y e. CC |-> ( x x. y ) )
4	3	fnmpo 6255	. . 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 2265	. . . . . . 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 1179	. . . 4 |- ( ( ( x e. CC /\ y e. CC ) /\ z = ( x x. y ) ) -> ( x e. CC /\ y e. CC /\ z e. CC ) )
12	11	ssoprab2i 6007	. . 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 5923	. . 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 6247	. . 3 |- ( ( CC X. CC ) X. CC ) = { <. <. x , y >. , z >. | ( x e. CC /\ y e. CC /\ z e. CC ) }
15	12, 13, 14	3sstr4i 3220	. 2 |- ( x e. CC , y e. CC |-> ( x x. y ) ) C_ ( ( CC X. CC ) X. CC )
16		dff2 5702	. 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 944	1 |- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3153   X. cxp 4657   Fn wfn 5249   -->wf 5250  (class class class)co 5918   {coprab 5919   e. cmpo 5920   CCcc 7870   x. cmul 7877

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-mulcl 7970

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194

This theorem is referenced by: (None)