Theorem addcnsr 4047

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 1893	. 2 |- <.(A +R C), (B +R D)>. e. V
2		opeq12 1878	. . . 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 3006	. . . 4 |- (w = A -> (w +R u) = (A +R u))
4		opreq1 3006	. . . 4 |- (v = B -> (v +R f) = (B +R f))
5	2, 3, 4	syl2an 349	. . 3 |- ((w = A /\ v = B) -> <.(w +R u), (v +R f)>. = <.(A +R u), (B +R f)>.)
6		opeq12 1878	. . . 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 3007	. . . 4 |- (u = C -> (A +R u) = (A +R C))
8		opreq2 3007	. . . 4 |- (f = D -> (B +R f) = (B +R D))
9	6, 7, 8	syl2an 349	. . 3 |- ((u = C /\ f = D) -> <.(A +R u), (B +R f)>. = <.(A +R C), (B +R D)>.)
10	5, 9	sylan9eq 1144	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.(w +R u), (v +R f)>. = <.(A +R C), (B +R D)>.)
11		df-plus 4039	. . 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 4034	. . . . . . 7 |- CC = (R. X. R.)
13	12	eleq2i 1153	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
14	12	eleq2i 1153	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
15	13, 14	anbi12i 369	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
16	15	anbi1i 368	. . . 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	bioprabi 3027	. . 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 1119	. 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 3052	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 2   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  <.cop 1810   X. cxp 2408  (class class class)co 3001  {copab2 3002  R.cnr 3787   +R cplr 3791  CCcc 4026   + caddc 4031

This theorem is referenced by:  addresr 4050  addcnsrec 4057  axaddcl 4066  ax0id 4076  axnegex 4078  axcnre 4087

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004  df-c 4034  df-plus 4039

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