Theorem ltsrpr 5158

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 5150	. 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 5147	. 2 |- dom ~R = (P. X. P.)
6		df-nr 5139	. 2 |- R. = ((P. X. P.)/. ~R )
7		ltrelsr 5152	. 2 |- <R (_ (R. X. R.)
8		ltrelpr 5073	. 2 |- <P (_ (P. X. P.)
9		0npr 5068	. 2 |- -. (/) e. P.
10		dmplp 5087	. 2 |- dom +P. = (P. X. P.)
11		enrer 5148	. . 3 |- Er ~R
12		df-ltr 5142	. . 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 5149	. . . . . 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 5149	. . . . . . 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 1469	. . . . . . 7 |- ((v +P. D) = (u +P. C) <-> (u +P. C) = (v +P. D))
16	14, 15	syl6bb 534	. . . . . 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 630	. . . . 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 3955	. . . . . 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 1804	. . . . . . 7 |- z e. V
20		visset 1804	. . . . . . 7 |- u e. V
21		visset 1804	. . . . . . . 8 |- x e. V
22		visset 1804	. . . . . . . 8 |- y e. V
23	21, 22	addcompr 5095	. . . . . . 7 |- (x +P. y) = (y +P. x)
24		visset 1804	. . . . . . . 8 |- f e. V
25	22, 24	addasspr 5096	. . . . . . 7 |- ((x +P. y) +P. f) = (x +P. (y +P. f))
26	19, 20, 2, 23, 25, 3	caopr4 4050	. . . . . 6 |- ((z +P. u) +P. (B +P. C)) = ((z +P. B) +P. (u +P. C))
27		visset 1804	. . . . . . 7 |- w e. V
28		visset 1804	. . . . . . 7 |- v e. V
29		ltsrpr.1	. . . . . . 7 |- A e. V
30	27, 28, 29, 23, 25, 4	caopr4 4050	. . . . . 6 |- ((w +P. v) +P. (A +P. D)) = ((w +P. A) +P. (v +P. D))
31	18, 26, 30	3eqtr4g 1523	. . . . 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 5092	. . . . . . . . 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 5092	. . . . . . . . 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 507	. . . . 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 3968	. . . . . . 7 |- (z +P. u) e. V
41		oprex 3968	. . . . . . 7 |- (B +P. C) e. V
42	21, 22	ltapr 5123	. . . . . . 7 |- (f e. P. -> (x <P y <-> (f +P. x) <P (f +P. y)))
43		oprex 3968	. . . . . . 7 |- (w +P. v) e. V
44		oprex 3968	. . . . . . 7 |- (A +P. D) e. V
45	40, 41, 42, 43, 23, 44	caoprord3 4044	. . . . . 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 4290	. 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 4291	1 |- ([<.A, B>.] ~R <R [<.C, D>.] ~R <-> (A +P. D) <P (B +P. C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802  <.cop 2401   class class class wbr 2609  (class class class)co 3948  [cec 4243  P.cnp 4957   +P. cpp 4959   <P cltp 4961   ~R cer 4964  R.cnr 4965   <R cltr 4971

This theorem is referenced by:  gt0srpr 5159  ltsosr 5175  0lt1sr 5176  ltasr 5181  mappsrpr 5190  ltpsrpr 5191  map2psrpr 5192

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-plp 5060  df-ltp 5062  df-enr 5138  df-nr 5139  df-ltr 5142

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