Theorem addcnsr 5225

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

Assertion

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

Proof of Theorem addcnsr

Step	Hyp	Ref	Expression
1		opex 2772	. 2 |- <.(A +R C), (B +R D)>. e. V
2		opeq12 2480	. . . 4 |- (((w +R u) = (A +R u) /\ (v +R f) = (B +R f)) -> <.(w +R u), (v +R f)>. = <.(A +R u), (B +R f)>.)
3		opreq1 3953	. . . 4 |- (w = A -> (w +R u) = (A +R u))
4		opreq1 3953	. . . 4 |- (v = B -> (v +R f) = (B +R f))
5	2, 3, 4	syl2an 454	. . 3 |- ((w = A /\ v = B) -> <.(w +R u), (v +R f)>. = <.(A +R u), (B +R f)>.)
6		opeq12 2480	. . . 4 |- (((A +R u) = (A +R C) /\ (B +R f) = (B +R D)) -> <.(A +R u), (B +R f)>. = <.(A +R C), (B +R D)>.)
7		opreq2 3954	. . . 4 |- (u = C -> (A +R u) = (A +R C))
8		opreq2 3954	. . . 4 |- (f = D -> (B +R f) = (B +R D))
9	6, 7, 8	syl2an 454	. . 3 |- ((u = C /\ f = D) -> <.(A +R u), (B +R f)>. = <.(A +R C), (B +R D)>.)
10	5, 9	sylan9eq 1519	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.(w +R u), (v +R f)>. = <.(A +R C), (B +R D)>.)
11		df-plus 5217	. . 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), (v +R f)>.))}
12		df-c 5212	. . . . . . 7 |- CC = (R. X. R.)
13	12	eleq2i 1530	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
14	12	eleq2i 1530	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
15	13, 14	anbi12i 481	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
16	15	anbi1i 480	. . . 4 |- (((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +R u), (v +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), (v +R f)>.)))
17	16	oprabbii 3982	. . 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), (v +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), (v +R f)>.))}
18	11, 17	eqtr 1487	. 2 |- + = {<.<.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), (v +R f)>.))}
19	1, 10, 18	oprabval3 4015	1 |- (((A e. R. /\ B e. R.) /\ (C e. R. /\ D e. R.)) -> (<.A, B>. + <.C, D>.) = <.(A +R C), (B +R D)>.)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  <.cop 2401   X. cxp 3158  (class class class)co 3948  {copab2 3949  R.cnr 4965   +R cplr 4969  CCcc 5204   + caddc 5209

This theorem is referenced by:  addresr 5228  addcnsrec 5235  axaddopr 5237  ax0id 5253  axcnre 5258

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-opr 3950  df-oprab 3951  df-c 5212  df-plus 5217

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