Theorem addcnsr 6405

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 3527	. 2 |- <.(A +R C), (B +R D)>. e. _V
2		opeq12 3160	. . . 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 4889	. . . 4 |- (w = A -> (w +R u) = (A +R u))
4		opreq1 4889	. . . 4 |- (v = B -> (v +R f) = (B +R f))
5	2, 3, 4	syl2an 503	. . 3 |- ((w = A /\ v = B) -> <.(w +R u), (v +R f)>. = <.(A +R u), (B +R f)>.)
6		opeq12 3160	. . . 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 4890	. . . 4 |- (u = C -> (A +R u) = (A +R C))
8		opreq2 4890	. . . 4 |- (f = D -> (B +R f) = (B +R D))
9	6, 7, 8	syl2an 503	. . 3 |- ((u = C /\ f = D) -> <.(A +R u), (B +R f)>. = <.(A +R C), (B +R D)>.)
10	5, 9	sylan9eq 1948	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.(w +R u), (v +R f)>. = <.(A +R C), (B +R D)>.)
11		df-plus 6397	. . 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 6392	. . . . . . 7 |- CC = (R. X. R.)
13	12	eleq2i 1961	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
14	12	eleq2i 1961	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
15	13, 14	anbi12i 540	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
16	15	anbi1i 539	. . . 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 4923	. . 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	eqtri 1908	. 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 4959	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 240   = wceq 1298   e. wcel 1300  E.wex 1326  <.cop 3046   X. cxp 3984  (class class class)co 4884  {copab2 4885  R.cnr 6145   +R cplr 6149  CCcc 6384   + caddc 6389

This theorem is referenced by:  addresr 6408  addcnsrec 6415  axaddopr 6417  ax0id 6434  axcnre 6439

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-c 6392  df-plus 6397

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