Theorem mulcnsr 5266

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 2788	. 2 |- <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>. e. V
2		opreq1 3974	. . . . 5 |- (w = A -> (w .R u) = (A .R u))
3		opreq1 3974	. . . . . 6 |- (v = B -> (v .R f) = (B .R f))
4	3	opreq2d 3982	. . . . 5 |- (v = B -> (-1R .R (v .R f)) = (-1R .R (B .R f)))
5	2, 4	opreqan12d 3985	. . . 4 |- ((w = A /\ v = B) -> ((w .R u) +R (-1R .R (v .R f))) = ((A .R u) +R (-1R .R (B .R f))))
6		opreq1 3974	. . . . 5 |- (v = B -> (v .R u) = (B .R u))
7		opreq1 3974	. . . . 5 |- (w = A -> (w .R f) = (A .R f))
8	6, 7	opreqan12rd 3986	. . . 4 |- ((w = A /\ v = B) -> ((v .R u) +R (w .R f)) = ((B .R u) +R (A .R f)))
9	5, 8	opeq12d 2499	. . 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))>.)
10		opreq2 3975	. . . . 5 |- (u = C -> (A .R u) = (A .R C))
11		opreq2 3975	. . . . . 6 |- (f = D -> (B .R f) = (B .R D))
12	11	opreq2d 3982	. . . . 5 |- (f = D -> (-1R .R (B .R f)) = (-1R .R (B .R D)))
13	10, 12	opreqan12d 3985	. . . 4 |- ((u = C /\ f = D) -> ((A .R u) +R (-1R .R (B .R f))) = ((A .R C) +R (-1R .R (B .R D))))
14		opreq2 3975	. . . . 5 |- (u = C -> (B .R u) = (B .R C))
15		opreq2 3975	. . . . 5 |- (f = D -> (A .R f) = (A .R D))
16	14, 15	opreqan12d 3985	. . . 4 |- ((u = C /\ f = D) -> ((B .R u) +R (A .R f)) = ((B .R C) +R (A .R D)))
17	13, 16	opeq12d 2499	. . 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))>.)
18	9, 17	sylan9eq 1530	. 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))>.)
19		df-mul 5258	. . 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))>.))}
20		df-c 5252	. . . . . . 7 |- CC = (R. X. R.)
21	20	eleq2i 1541	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
22	20	eleq2i 1541	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
23	21, 22	anbi12i 484	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
24	23	anbi1i 483	. . . 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))>.)))
25	24	oprabbii 4003	. . 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))>.))}
26	19, 25	eqtr 1498	. 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))>.))}
27	1, 18, 26	oprabval3 4036	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 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  <.cop 2415   X. cxp 3174  (class class class)co 3969  {copab2 3970  R.cnr 5005  -1Rcm1r 5008   +R cplr 5009   .R cmr 5010  CCcc 5244   x. cmul 5251

This theorem is referenced by:  mulresr 5269  mulcnsrec 5276  axmulopr 5278  ax1id 5294  axi2m1 5297  axcnre 5298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-oprab 3972  df-c 5252  df-mul 5258

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