Theorem ltsrpr 5186

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

Hypotheses

Ref	Expression
ltsrpr.1	|- A e. V
ltsrpr.2	|- B e. V
ltsrpr.3	|- C e. V
ltsrpr.4	|- D e. V

Assertion

Ref	Expression
ltsrpr	|- ([<.A, B>.] ~R <R [<.C, D>.] ~R <-> (A +P. D) <P (B +P. C))

Proof of Theorem ltsrpr

Step	Hyp	Ref	Expression
1		enrex 5178	. 2 |- ~R e. V
2		ltsrpr.2	. 2 |- B e. V
3		ltsrpr.3	. 2 |- C e. V
4		ltsrpr.4	. 2 |- D e. V
5		dmenr 5175	. 2 |- dom ~R = (P. X. P.)
6		df-nr 5167	. 2 |- R. = ((P. X. P.)/. ~R )
7		ltrelsr 5180	. 2 |- <R (_ (R. X. R.)
8		ltrelpr 5101	. 2 |- <P (_ (P. X. P.)
9		0npr 5096	. 2 |- -. (/) e. P.
10		dmplp 5115	. 2 |- dom +P. = (P. X. P.)
11		enrer 5176	. . 3 |- Er ~R
12		df-ltr 5170	. . 3 |- <R = {<.x, y>. | ((x e. R. /\ y e. R.) /\ E.zE.wE.vE.u((x = [<.z, w>.] ~R /\ y = [<.v, u>.] ~R ) /\ (z +P. u) <P (w +P. v)))}
13		enreceq 5177	. . . . . 6 |- (((z e. P. /\ w e. P.) /\ (A e. P. /\ B e. P.)) -> ([<.z, w>.] ~R = [<.A, B>.] ~R <-> (z +P. B) = (w +P. A)))
14		enreceq 5177	. . . . . . 7 |- (((v e. P. /\ u e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.v, u>.] ~R = [<.C, D>.] ~R <-> (v +P. D) = (u +P. C)))
15		eqcom 1477	. . . . . . 7 |- ((v +P. D) = (u +P. C) <-> (u +P. C) = (v +P. D))
16	14, 15	syl6bb 536	. . . . . 6 |- (((v e. P. /\ u e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.v, u>.] ~R = [<.C, D>.] ~R <-> (u +P. C) = (v +P. D)))
17	13, 16	bi2anan9 632	. . . . 5 |- ((((z e. P. /\ w e. P.) /\ (A e. P. /\ B e. P.)) /\ ((v e. P. /\ u e. P.) /\ (C e. P. /\ D e. P.))) -> (([<.z, w>.] ~R = [<.A, B>.] ~R /\ [<.v, u>.] ~R = [<.C, D>.] ~R ) <-> ((z +P. B) = (w +P. A) /\ (u +P. C) = (v +P. D))))
18		opreq12 3970	. . . . . 6 |- (((z +P. B) = (w +P. A) /\ (u +P. C) = (v +P. D)) -> ((z +P. B) +P. (u +P. C)) = ((w +P. A) +P. (v +P. D)))
19		visset 1813	. . . . . . 7 |- z e. V
20		visset 1813	. . . . . . 7 |- u e. V
21		visset 1813	. . . . . . . 8 |- x e. V
22		visset 1813	. . . . . . . 8 |- y e. V
23	21, 22	addcompr 5123	. . . . . . 7 |- (x +P. y) = (y +P. x)
24		visset 1813	. . . . . . . 8 |- f e. V
25	22, 24	addasspr 5124	. . . . . . 7 |- ((x +P. y) +P. f) = (x +P. (y +P. f))
26	19, 20, 2, 23, 25, 3	caopr4 4064	. . . . . 6 |- ((z +P. u) +P. (B +P. C)) = ((z +P. B) +P. (u +P. C))
27		visset 1813	. . . . . . 7 |- w e. V
28		visset 1813	. . . . . . 7 |- v e. V
29		ltsrpr.1	. . . . . . 7 |- A e. V
30	27, 28, 29, 23, 25, 4	caopr4 4064	. . . . . 6 |- ((w +P. v) +P. (A +P. D)) = ((w +P. A) +P. (v +P. D))
31	18, 26, 30	3eqtr4g 1531	. . . . 5 |- (((z +P. B) = (w +P. A) /\ (u +P. C) = (v +P. D)) -> ((z +P. u) +P. (B +P. C)) = ((w +P. v) +P. (A +P. D)))
32	17, 31	syl6bi 214	. . . 4 |- ((((z e. P. /\ w e. P.) /\ (A e. P. /\ B e. P.)) /\ ((v e. P. /\ u e. P.) /\ (C e. P. /\ D e. P.))) -> (([<.z, w>.] ~R = [<.A, B>.] ~R /\ [<.v, u>.] ~R = [<.C, D>.] ~R ) -> ((z +P. u) +P. (B +P. C)) = ((w +P. v) +P. (A +P. D))))
33		addclpr 5120	. . . . . . . . 9 |- ((B e. P. /\ C e. P.) -> (B +P. C) e. P.)
34	33	ad2ant2lr 410	. . . . . . . 8 |- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> (B +P. C) e. P.)
35		addclpr 5120	. . . . . . . . 9 |- ((w e. P. /\ v e. P.) -> (w +P. v) e. P.)
36	35	ad2ant2lr 410	. . . . . . . 8 |- (((z e. P. /\ w e. P.) /\ (v e. P. /\ u e. P.)) -> (w +P. v) e. P.)
37	34, 36	anim12i 333	. . . . . . 7 |- ((((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) /\ ((z e. P. /\ w e. P.) /\ (v e. P. /\ u e. P.))) -> ((B +P. C) e. P. /\ (w +P. v) e. P.))
38	37	ancoms 436	. . . . . 6 |- ((((z e. P. /\ w e. P.) /\ (v e. P. /\ u e. P.)) /\ ((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.))) -> ((B +P. C) e. P. /\ (w +P. v) e. P.))
39	38	an4s 508	. . . . 5 |- ((((z e. P. /\ w e. P.) /\ (A e. P. /\ B e. P.)) /\ ((v e. P. /\ u e. P.) /\ (C e. P. /\ D e. P.))) -> ((B +P. C) e. P. /\ (w +P. v) e. P.))
40		oprex 3983	. . . . . . 7 |- (z +P. u) e. V
41		oprex 3983	. . . . . . 7 |- (B +P. C) e. V
42	21, 22	ltapr 5151	. . . . . . 7 |- (f e. P. -> (x <P y <-> (f +P. x) <P (f +P. y)))
43		oprex 3983	. . . . . . 7 |- (w +P. v) e. V
44		oprex 3983	. . . . . . 7 |- (A +P. D) e. V
45	40, 41, 42, 43, 23, 44	caoprord3 4058	. . . . . 6 |- ((((B +P. C) e. P. /\ (w +P. v) e. P.) /\ ((z +P. u) +P. (B +P. C)) = ((w +P. v) +P. (A +P. D))) -> ((z +P. u) <P (w +P. v) <-> (A +P. D) <P (B +P. C)))
46	45	ex 373	. . . . 5 |- (((B +P. C) e. P. /\ (w +P. v) e. P.) -> (((z +P. u) +P. (B +P. C)) = ((w +P. v) +P. (A +P. D)) -> ((z +P. u) <P (w +P. v) <-> (A +P. D) <P (B +P. C))))
47	39, 46	syl 10	. . . 4 |- ((((z e. P. /\ w e. P.) /\ (A e. P. /\ B e. P.)) /\ ((v e. P. /\ u e. P.) /\ (C e. P. /\ D e. P.))) -> (((z +P. u) +P. (B +P. C)) = ((w +P. v) +P. (A +P. D)) -> ((z +P. u) <P (w +P. v) <-> (A +P. D) <P (B +P. C))))
48	32, 47	syld 27	. . 3 |- ((((z e. P. /\ w e. P.) /\ (A e. P. /\ B e. P.)) /\ ((v e. P. /\ u e. P.) /\ (C e. P. /\ D e. P.))) -> (([<.z, w>.] ~R = [<.A, B>.] ~R /\ [<.v, u>.] ~R = [<.C, D>.] ~R ) -> ((z +P. u) <P (w +P. v) <-> (A +P. D) <P (B +P. C))))
49	1, 11, 5, 6, 12, 48	brecop 4306	. 2 |- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.A, B>.] ~R <R [<.C, D>.] ~R <-> (A +P. D) <P (B +P. C)))
50	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 49	brecop2 4307	1 |- ([<.A, B>.] ~R <R [<.C, D>.] ~R <-> (A +P. D) <P (B +P. C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  <.cop 2411   class class class wbr 2619  (class class class)co 3963  [cec 4259  P.cnp 4985   +P. cpp 4987   <P cltp 4989   ~R cer 4992  R.cnr 4993   <R cltr 4999

This theorem is referenced by:  gt0srpr 5187  ltsosr 5203  0lt1sr 5204  ltasr 5209  mappsrpr 5218  ltpsrpr 5219  map2psrpr 5220

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-plp 5088  df-ltp 5090  df-enr 5166  df-nr 5167  df-ltr 5170

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