Theorem addsrpr 3978

Description: Addition of signed reals in terms of positive reals.

Assertion

Ref	Expression
addsrpr	|- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.A, B>.] ~R +R [<.C, D>.] ~R ) = [<.(A +P. C), (B +P. D)>.] ~R )

Proof of Theorem addsrpr

Step	Hyp	Ref	Expression
1		opex 1893	. 2 |- <.(A +P. C), (B +P. D)>. e. V
2		opex 1893	. 2 |- <.(a +P. g), (b +P. h)>. e. V
3		opex 1893	. 2 |- <.(c +P. t), (d +P. s)>. e. V
4		enrex 3972	. 2 |- ~R e. V
5		enrer 3970	. 2 |- Er ~R
6		dmenr 3969	. 2 |- dom ~R = (P. X. P.)
7		df-enr 3960	. 2 |- ~R = {<.x, y>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (z +P. u) = (w +P. v)))}
8		opreq12 3008	. . . 4 |- ((z = a /\ u = d) -> (z +P. u) = (a +P. d))
9		opreq12 3008	. . . 4 |- ((w = b /\ v = c) -> (w +P. v) = (b +P. c))
10	8, 9	cleqan12d 1116	. . 3 |- (((z = a /\ u = d) /\ (w = b /\ v = c)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
11	10	an42s 391	. 2 |- (((z = a /\ w = b) /\ (v = c /\ u = d)) -> ((z +P. u) = (w +P. v) <-> (a +P. d) = (b +P. c)))
12		opreq12 3008	. . . 4 |- ((z = g /\ u = s) -> (z +P. u) = (g +P. s))
13		opreq12 3008	. . . 4 |- ((w = h /\ v = t) -> (w +P. v) = (h +P. t))
14	12, 13	cleqan12d 1116	. . 3 |- (((z = g /\ u = s) /\ (w = h /\ v = t)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
15	14	an42s 391	. 2 |- (((z = g /\ w = h) /\ (v = t /\ u = s)) -> ((z +P. u) = (w +P. v) <-> (g +P. s) = (h +P. t)))
16		df-plpr 3958	. 2 |- +pR = {<.<.x, y>., z>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.))}
17		opeq12 1878	. . . 4 |- (((w +P. u) = (a +P. g) /\ (v +P. f) = (b +P. h)) -> <.(w +P. u), (v +P. f)>. = <.(a +P. g), (b +P. h)>.)
18		opreq12 3008	. . . 4 |- ((w = a /\ u = g) -> (w +P. u) = (a +P. g))
19		opreq12 3008	. . . 4 |- ((v = b /\ f = h) -> (v +P. f) = (b +P. h))
20	17, 18, 19	syl2an 349	. . 3 |- (((w = a /\ u = g) /\ (v = b /\ f = h)) -> <.(w +P. u), (v +P. f)>. = <.(a +P. g), (b +P. h)>.)
21	20	an4s 390	. 2 |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> <.(w +P. u), (v +P. f)>. = <.(a +P. g), (b +P. h)>.)
22		opeq12 1878	. . . 4 |- (((w +P. u) = (c +P. t) /\ (v +P. f) = (d +P. s)) -> <.(w +P. u), (v +P. f)>. = <.(c +P. t), (d +P. s)>.)
23		opreq12 3008	. . . 4 |- ((w = c /\ u = t) -> (w +P. u) = (c +P. t))
24		opreq12 3008	. . . 4 |- ((v = d /\ f = s) -> (v +P. f) = (d +P. s))
25	22, 23, 24	syl2an 349	. . 3 |- (((w = c /\ u = t) /\ (v = d /\ f = s)) -> <.(w +P. u), (v +P. f)>. = <.(c +P. t), (d +P. s)>.)
26	25	an4s 390	. 2 |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> <.(w +P. u), (v +P. f)>. = <.(c +P. t), (d +P. s)>.)
27		opeq12 1878	. . . 4 |- (((w +P. u) = (A +P. C) /\ (v +P. f) = (B +P. D)) -> <.(w +P. u), (v +P. f)>. = <.(A +P. C), (B +P. D)>.)
28		opreq12 3008	. . . 4 |- ((w = A /\ u = C) -> (w +P. u) = (A +P. C))
29		opreq12 3008	. . . 4 |- ((v = B /\ f = D) -> (v +P. f) = (B +P. D))
30	27, 28, 29	syl2an 349	. . 3 |- (((w = A /\ u = C) /\ (v = B /\ f = D)) -> <.(w +P. u), (v +P. f)>. = <.(A +P. C), (B +P. D)>.)
31	30	an4s 390	. 2 |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> <.(w +P. u), (v +P. f)>. = <.(A +P. C), (B +P. D)>.)
32		df-plr 3962	. 2 |- +R = {<.<.x, y>., z>. | ((x e. R. /\ y e. R.) /\ E.aE.bE.cE.d((x = [<.a, b>.] ~R /\ y = [<.c, d>.] ~R ) /\ z = [(<.a, b>. +pR <.c, d>.)] ~R ))}
33		df-nr 3961	. 2 |- R. = ((P. X. P.)/. ~R )
34		visset 1350	. . 3 |- a e. V
35		visset 1350	. . 3 |- b e. V
36		visset 1350	. . 3 |- c e. V
37		visset 1350	. . 3 |- d e. V
38		visset 1350	. . 3 |- g e. V
39		visset 1350	. . 3 |- h e. V
40		visset 1350	. . 3 |- t e. V
41		visset 1350	. . 3 |- s e. V
42	34, 35, 36, 37, 38, 39, 40, 41	addcmpblnr 3975	. 2 |- ((((a e. P. /\ b e. P.) /\ (c e. P. /\ d e. P.)) /\ ((g e. P. /\ h e. P.) /\ (t e. P. /\ s e. P.))) -> (((a +P. d) = (b +P. c) /\ (g +P. s) = (h +P. t)) -> <.(a +P. g), (b +P. h)>. ~R <.(c +P. t), (d +P. s)>.))
43	1, 2, 3, 4, 5, 6, 7, 11, 15, 16, 21, 26, 31, 32, 33, 42	oprec 3254	1 |- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.A, B>.] ~R +R [<.C, D>.] ~R ) = [<.(A +P. C), (B +P. D)>.] ~R )

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  <.cop 1810  (class class class)co 3001  [cec 3198  P.cnp 3779   +P. cpp 3781   +pR cplpr 3784   ~R cer 3786  R.cnr 3787   +R cplr 3791

This theorem is referenced by:  addclsr 3986  addcomsr 3990  addasssr 3991  distrsr 3994  m1p1sr 3995  0idsr 4000  ltasr 4003

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-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  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-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  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-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962

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