Theorem ltsrpr 11037

Description: Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
ltsrpr	⊢ ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))

Proof of Theorem ltsrpr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrer 11023	. . 3 ⊢ ~R Er (P × P)
2		erdm 8684	. . 3 ⊢ ( ~R Er (P × P) → dom ~R = (P × P))
3	1, 2	ax-mp 5	. 2 ⊢ dom ~R = (P × P)
4		df-nr 11016	. 2 ⊢ R = ((P × P) / ~R )
5		ltrelsr 11028	. 2 ⊢ <R ⊆ (R × R)
6		ltrelpr 10958	. 2 ⊢ <P ⊆ (P × P)
7		0npr 10952	. 2 ⊢ ¬ ∅ ∈ P
8		dmplp 10972	. 2 ⊢ dom +P = (P × P)
9		enrex 11027	. . 3 ⊢ ~R ∈ V
10		df-ltr 11019	. . 3 ⊢ <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
11		addclpr 10978	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 +P 𝑣) ∈ P)
12	11	ad2ant2lr 748	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑤 +P 𝑣) ∈ P)
13		addclpr 10978	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
14	13	ad2ant2lr 748	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐵 +P 𝐶) ∈ P)
15	12, 14	anim12ci 614	. . . . 5 ⊢ ((((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
16	15	an4s 660	. . . 4 ⊢ ((((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) ∧ ((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
17		enreceq 11026	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18		enreceq 11026	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19		eqcom 2737	. . . . . . 7 ⊢ ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
20	18, 19	bitrdi 287	. . . . . 6 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷)))
21	17, 20	bi2anan9 638	. . . . 5 ⊢ ((((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) ∧ ((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ 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))))
22		oveq12 7399	. . . . . 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)))
23		addcompr 10981	. . . . . . . . . 10 ⊢ (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
24	23	oveq1i 7400	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25		addasspr 10982	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26		addasspr 10982	. . . . . . . . 9 ⊢ ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
27	24, 25, 26	3eqtr3i 2761	. . . . . . . 8 ⊢ (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
28	27	oveq2i 7401	. . . . . . 7 ⊢ (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29		addasspr 10982	. . . . . . 7 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30		addasspr 10982	. . . . . . 7 ⊢ ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
31	28, 29, 30	3eqtr4i 2763	. . . . . 6 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32		addcompr 10981	. . . . . . . . . 10 ⊢ (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
33	32	oveq1i 7400	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34		addasspr 10982	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35		addasspr 10982	. . . . . . . . 9 ⊢ ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
36	33, 34, 35	3eqtr3i 2761	. . . . . . . 8 ⊢ (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
37	36	oveq2i 7401	. . . . . . 7 ⊢ (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38		addasspr 10982	. . . . . . 7 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39		addasspr 10982	. . . . . . 7 ⊢ ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
40	37, 38, 39	3eqtr4i 2763	. . . . . 6 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
41	22, 31, 40	3eqtr4g 2790	. . . . 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)))
42	21, 41	biimtrdi 253	. . . 4 ⊢ ((((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) ∧ ((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ 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))))
43		ovex 7423	. . . . 5 ⊢ (𝑧 +P 𝑢) ∈ V
44		ovex 7423	. . . . 5 ⊢ (𝐵 +P 𝐶) ∈ V
45		ltapr 11005	. . . . 5 ⊢ (𝑓 ∈ P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46		ovex 7423	. . . . 5 ⊢ (𝑤 +P 𝑣) ∈ V
47		addcompr 10981	. . . . 5 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48		ovex 7423	. . . . 5 ⊢ (𝐴 +P 𝐷) ∈ V
49	43, 44, 45, 46, 47, 48	caovord3 7605	. . . 4 ⊢ ((((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ 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)))
50	16, 42, 49	syl6an 684	. . 3 ⊢ ((((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) ∧ ((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ 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))))
51	9, 1, 4, 10, 50	brecop 8786	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))
52	3, 4, 5, 6, 7, 8, 51	brecop2 8787	1 ⊢ ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   class class class wbr 5110   × cxp 5639  dom cdm 5641  (class class class)co 7390   Er wer 8671  [cec 8672  Pcnp 10819   +_P cpp 10821  <_P cltp 10823   ~_R cer 10824  Rcnr 10825   <_R cltr 10831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8674  df-ec 8676  df-qs 8680  df-ni 10832  df-pli 10833  df-mi 10834  df-lti 10835  df-plpq 10868  df-mpq 10869  df-ltpq 10870  df-enq 10871  df-nq 10872  df-erq 10873  df-plq 10874  df-mq 10875  df-1nq 10876  df-rq 10877  df-ltnq 10878  df-np 10941  df-plp 10943  df-ltp 10945  df-enr 11015  df-nr 11016  df-ltr 11019

This theorem is referenced by:  gt0srpr  11038  ltsosr  11054  0lt1sr  11055  ltasr  11060  mappsrpr  11068  ltpsrpr  11069  map2psrpr  11070