Theorem mulcnsr 4048

Description: Multiplication of complex numbers in terms of signed reals.

Assertion

Ref	Expression
mulcnsr	|- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> (<.A, B>. x. <.C, D>.) = <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.)

Proof of Theorem mulcnsr

Step	Hyp	Ref	Expression
1		opex 1893	. 2 |- <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>. e. V
2		opeq12 1878	. . . 4 |- ((((w .R u) +R (-1R .R (v .R f))) = ((A .R u) +R (-1R .R (B .R f))) /\ ((v .R u) +R (w .R f)) = ((B .R u) +R (A .R f))) -> <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>. = <.((A .R u) +R (-1R .R (B .R f))), ((B .R u) +R (A .R f))>.)
3		opreq1 3006	. . . . 5 |- (w = A -> (w .R u) = (A .R u))
4		opreq1 3006	. . . . . 6 |- (v = B -> (v .R f) = (B .R f))
5	4	opreq2d 3013	. . . . 5 |- (v = B -> (-1R .R (v .R f)) = (-1R .R (B .R f)))
6	3, 5	opreqan12d 3015	. . . 4 |- ((w = A /\ v = B) -> ((w .R u) +R (-1R .R (v .R f))) = ((A .R u) +R (-1R .R (B .R f))))
7		opreq1 3006	. . . . 5 |- (v = B -> (v .R u) = (B .R u))
8		opreq1 3006	. . . . 5 |- (w = A -> (w .R f) = (A .R f))
9	7, 8	opreqan12rd 3016	. . . 4 |- ((w = A /\ v = B) -> ((v .R u) +R (w .R f)) = ((B .R u) +R (A .R f)))
10	2, 6, 9	sylanc 361	. . 3 |- ((w = A /\ v = B) -> <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>. = <.((A .R u) +R (-1R .R (B .R f))), ((B .R u) +R (A .R f))>.)
11		opeq12 1878	. . . 4 |- ((((A .R u) +R (-1R .R (B .R f))) = ((A .R C) +R (-1R .R (B .R D))) /\ ((B .R u) +R (A .R f)) = ((B .R C) +R (A .R D))) -> <.((A .R u) +R (-1R .R (B .R f))), ((B .R u) +R (A .R f))>. = <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.)
12		opreq2 3007	. . . . 5 |- (u = C -> (A .R u) = (A .R C))
13		opreq2 3007	. . . . . 6 |- (f = D -> (B .R f) = (B .R D))
14	13	opreq2d 3013	. . . . 5 |- (f = D -> (-1R .R (B .R f)) = (-1R .R (B .R D)))
15	12, 14	opreqan12d 3015	. . . 4 |- ((u = C /\ f = D) -> ((A .R u) +R (-1R .R (B .R f))) = ((A .R C) +R (-1R .R (B .R D))))
16		opreq2 3007	. . . . 5 |- (u = C -> (B .R u) = (B .R C))
17		opreq2 3007	. . . . 5 |- (f = D -> (A .R f) = (A .R D))
18	16, 17	opreqan12d 3015	. . . 4 |- ((u = C /\ f = D) -> ((B .R u) +R (A .R f)) = ((B .R C) +R (A .R D)))
19	11, 15, 18	sylanc 361	. . 3 |- ((u = C /\ f = D) -> <.((A .R u) +R (-1R .R (B .R f))), ((B .R u) +R (A .R f))>. = <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.)
20	10, 19	sylan9eq 1144	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>. = <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.)
21		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))>.))}
22		df-c 4034	. . . . . . 7 |- CC = (R. X. R.)
23	22	eleq2i 1153	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
24	22	eleq2i 1153	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
25	23, 24	anbi12i 369	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
26	25	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))>.)))
27	26	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))>.))}
28	21, 27	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))>.))}
29	1, 20, 28	oprabval3 3052	1 |- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> (<.A, B>. x. <.C, D>.) = <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>.)

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  <.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:  mulresr 4051  mulcnsrec 4058  axmulcl 4068  ax1id 4077  axrecex 4079  axi2m1 4082  axcnre 4087

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  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-fv 2438  df-opr 3003  df-oprab 3004  df-c 4034  df-mul 4040

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