Theorem axmulex 4060

Description: The multiplication operation is a set. Supplemental axiom for complex numbers (needed by seqsuc 4671). (It is not clear if this should be added to the "official" list of complex number axioms. df-seq 4661 is an artificial construct that in principle could be defined differently so as not to require this axiom. Currently we do not include this axiom on the Metamath Proof Explorer page with the Axioms for Complex Numbers.)

Assertion

Ref	Expression
axmulex	|- x. e. V

Proof of Theorem axmulex

Step	Hyp	Ref	Expression
1		srex 3973	. 2 |- R. e. V
2		df-mul 4040	. . 3 |- x. = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.))}
3		df-c 4034	. . . . . . 7 |- CC = (R. X. R.)
4	3	eleq2i 1153	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
5	3	eleq2i 1153	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
6	4, 5	anbi12i 369	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
7	6	anbi1i 368	. . . 4 |- (((x e. CC /\ y e. CC) /\ E.wE.vE.uE.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 e. (R. X. R.) /\ y e. (R. X. R.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.)))
8	7	bioprabi 3027	. . 3 |- {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.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, y>., z>. | ((x e. (R. X. R.) /\ y e. (R. X. R.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.))}
9	2, 8	eqtr 1119	. 2 |- x. = {<.<.x, y>., z>. | ((x e. (R. X. R.) /\ y e. (R. X. R.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.))}
10	1, 9	oprabex3 3046	1 |- x. e. V

Syntax hints:   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810   X. cxp 2408  (class class class)co 3001  {copab2 3002  R.cnr 3787  -1Rcm1r 3790   +R cplr 3791   .R cmr 3792  CCcc 4026   x. cmulc 4032

This theorem is referenced by:  expp1t 4678  exp1t 4679  fac0 4871  fac1 4872  facp1t 4873

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-oprab 3004  df-qs 3205  df-ni 3794  df-nq 3832  df-np 3880  df-nr 3961  df-c 4034  df-mul 4040

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