Theorem addcnsr 5265

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 2788	. 2 |- <.(A +R C), (B +R D)>. e. V
2		opeq12 2493	. . . 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 3974	. . . 4 |- (w = A -> (w +R u) = (A +R u))
4		opreq1 3974	. . . 4 |- (v = B -> (v +R f) = (B +R f))
5	2, 3, 4	syl2an 456	. . 3 |- ((w = A /\ v = B) -> <.(w +R u), (v +R f)>. = <.(A +R u), (B +R f)>.)
6		opeq12 2493	. . . 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 3975	. . . 4 |- (u = C -> (A +R u) = (A +R C))
8		opreq2 3975	. . . 4 |- (f = D -> (B +R f) = (B +R D))
9	6, 7, 8	syl2an 456	. . 3 |- ((u = C /\ f = D) -> <.(A +R u), (B +R f)>. = <.(A +R C), (B +R D)>.)
10	5, 9	sylan9eq 1530	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.(w +R u), (v +R f)>. = <.(A +R C), (B +R D)>.)
11		df-plus 5257	. . 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 5252	. . . . . . 7 |- CC = (R. X. R.)
13	12	eleq2i 1541	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
14	12	eleq2i 1541	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
15	13, 14	anbi12i 484	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
16	15	anbi1i 483	. . . 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 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), (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 1498	. 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 4036	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 958   e. wcel 960  E.wex 982  <.cop 2415   X. cxp 3174  (class class class)co 3969  {copab2 3970  R.cnr 5005   +R cplr 5009  CCcc 5244   + caddc 5249

This theorem is referenced by:  addresr 5268  addcnsrec 5275  axaddopr 5277  ax0id 5293  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-plus 5257

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