Theorem addcnsr 5318

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 2838	. 2 |- <.(A +R C), (B +R D)>. e. V
2		opeq12 2543	. . . 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 4026	. . . 4 |- (w = A -> (w +R u) = (A +R u))
4		opreq1 4026	. . . 4 |- (v = B -> (v +R f) = (B +R f))
5	2, 3, 4	syl2an 465	. . 3 |- ((w = A /\ v = B) -> <.(w +R u), (v +R f)>. = <.(A +R u), (B +R f)>.)
6		opeq12 2543	. . . 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 4027	. . . 4 |- (u = C -> (A +R u) = (A +R C))
8		opreq2 4027	. . . 4 |- (f = D -> (B +R f) = (B +R D))
9	6, 7, 8	syl2an 465	. . 3 |- ((u = C /\ f = D) -> <.(A +R u), (B +R f)>. = <.(A +R C), (B +R D)>.)
10	5, 9	sylan9eq 1574	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.(w +R u), (v +R f)>. = <.(A +R C), (B +R D)>.)
11		df-plus 5310	. . 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 5305	. . . . . . 7 |- CC = (R. X. R.)
13	12	eleq2i 1585	. . . . . 6 |- (x e. CC <-> x e. (R. X. R.))
14	12	eleq2i 1585	. . . . . 6 |- (y e. CC <-> y e. (R. X. R.))
15	13, 14	anbi12i 493	. . . . 5 |- ((x e. CC /\ y e. CC) <-> (x e. (R. X. R.) /\ y e. (R. X. R.)))
16	15	anbi1i 492	. . . 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 4055	. . 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 1542	. 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 4088	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 230   = wceq 997   e. wcel 999  E.wex 1021  <.cop 2463   X. cxp 3225  (class class class)co 4021  {copab2 4022  R.cnr 5058   +R cplr 5062  CCcc 5297   + caddc 5302

This theorem is referenced by:  addresr 5321  addcnsrec 5328  axaddopr 5330  ax0id 5346  axcnre 5351

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fv 3255  df-opr 4023  df-oprab 4024  df-c 5305  df-plus 5310

Copyright terms: Public domain