Theorem mulcnsr 5234

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 2777	. 2 |- <.((A .R C) +R (-1R .R (B .R D))), ((B .R C) +R (A .R D))>. e. V
2		opreq1 3959	. . . . 5 |- (w = A -> (w .R u) = (A .R u))
3		opreq1 3959	. . . . . 6 |- (v = B -> (v .R f) = (B .R f))
4	3	opreq2d 3967	. . . . 5 |- (v = B -> (-1R .R (v .R f)) = (-1R .R (B .R f)))
5	2, 4	opreqan12d 3970	. . . 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 3959	. . . . 5 |- (v = B -> (v .R u) = (B .R u))
7		opreq1 3959	. . . . 5 |- (w = A -> (w .R f) = (A .R f))
8	6, 7	opreqan12rd 3971	. . . 4 |- ((w = A /\ v = B) -> ((v .R u) +R (w .R f)) = ((B .R u) +R (A .R f)))
9	5, 8	opeq12d 2491	. . 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 3960	. . . . 5 |- (u = C -> (A .R u) = (A .R C))
11		opreq2 3960	. . . . . 6 |- (f = D -> (B .R f) = (B .R D))
12	11	opreq2d 3967	. . . . 5 |- (f = D -> (-1R .R (B .R f)) = (-1R .R (B .R D)))
13	10, 12	opreqan12d 3970	. . . 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 3960	. . . . 5 |- (u = C -> (B .R u) = (B .R C))
15		opreq2 3960	. . . . 5 |- (f = D -> (A .R f) = (A .R D))
16	14, 15	opreqan12d 3970	. . . 4 |- ((u = C /\ f = D) -> ((B .R u) +R (A .R f)) = ((B .R C) +R (A .R D)))
17	13, 16	opeq12d 2491	. . 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 1524	. 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 5226	. . 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 5220	. . . . . . 7 |- CC = (R. X. R.)
21	20	eleq2i 1535	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
22	20	eleq2i 1535	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
23	21, 22	anbi12i 482	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
24	23	anbi1i 481	. . . 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 3988	. . 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 1492	. 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 4021	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 954   e. wcel 956  E.wex 978  <.cop 2407   X. cxp 3163  (class class class)co 3954  {copab2 3955  R.cnr 4973  -1Rcm1r 4976   +R cplr 4977   .R cmr 4978  CCcc 5212   x. cmul 5219

This theorem is referenced by:  mulresr 5237  mulcnsrec 5244  axmulopr 5246  ax1id 5262  axi2m1 5265  axcnre 5266

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-opr 3956  df-oprab 3957  df-c 5220  df-mul 5226

Copyright terms: Public domain