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Theorem ltsrpr 10691
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.
StepHypRef Expression
1 enrer 10677 . . 3 ~R Er (P × P)
2 erdm 8401 . . 3 ( ~R Er (P × P) → dom ~R = (P × P))
31, 2ax-mp 5 . 2 dom ~R = (P × P)
4 df-nr 10670 . 2 R = ((P × P) / ~R )
5 ltrelsr 10682 . 2 <R ⊆ (R × R)
6 ltrelpr 10612 . 2 <P ⊆ (P × P)
7 0npr 10606 . 2 ¬ ∅ ∈ P
8 dmplp 10626 . 2 dom +P = (P × P)
9 enrex 10681 . . 3 ~R ∈ V
10 df-ltr 10673 . . 3 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
11 addclpr 10632 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 +P 𝑣) ∈ P)
1211ad2ant2lr 748 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑤 +P 𝑣) ∈ P)
13 addclpr 10632 . . . . . . 7 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
1413ad2ant2lr 748 . . . . . 6 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (𝐵 +P 𝐶) ∈ P)
1512, 14anim12ci 617 . . . . 5 ((((𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ ((𝐴P𝐵P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
1615an4s 660 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
17 enreceq 10680 . . . . . 6 (((𝑧P𝑤P) ∧ (𝐴P𝐵P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18 enreceq 10680 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19 eqcom 2744 . . . . . . 7 ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
2018, 19bitrdi 290 . . . . . 6 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷)))
2117, 20bi2anan9 639 . . . . 5 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (([⟨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 7222 . . . . . 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 10635 . . . . . . . . . 10 (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
2423oveq1i 7223 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25 addasspr 10636 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26 addasspr 10636 . . . . . . . . 9 ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
2724, 25, 263eqtr3i 2773 . . . . . . . 8 (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
2827oveq2i 7224 . . . . . . 7 (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29 addasspr 10636 . . . . . . 7 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30 addasspr 10636 . . . . . . 7 ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
3128, 29, 303eqtr4i 2775 . . . . . 6 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32 addcompr 10635 . . . . . . . . . 10 (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
3332oveq1i 7223 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34 addasspr 10636 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35 addasspr 10636 . . . . . . . . 9 ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
3633, 34, 353eqtr3i 2773 . . . . . . . 8 (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
3736oveq2i 7224 . . . . . . 7 (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38 addasspr 10636 . . . . . . 7 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39 addasspr 10636 . . . . . . 7 ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
4037, 38, 393eqtr4i 2775 . . . . . 6 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
4122, 31, 403eqtr4g 2803 . . . . 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)))
4221, 41syl6bi 256 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (([⟨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 7246 . . . . 5 (𝑧 +P 𝑢) ∈ V
44 ovex 7246 . . . . 5 (𝐵 +P 𝐶) ∈ V
45 ltapr 10659 . . . . 5 (𝑓P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46 ovex 7246 . . . . 5 (𝑤 +P 𝑣) ∈ V
47 addcompr 10635 . . . . 5 (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48 ovex 7246 . . . . 5 (𝐴 +P 𝐷) ∈ V
4943, 44, 45, 46, 47, 48caovord3 7421 . . . 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)))
5016, 42, 49syl6an 684 . . 3 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (([⟨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))))
519, 1, 4, 10, 50brecop 8492 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))
523, 4, 5, 6, 7, 8, 51brecop2 8493 1 ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2110  cop 4547   class class class wbr 5053   × cxp 5549  dom cdm 5551  (class class class)co 7213   Er wer 8388  [cec 8389  Pcnp 10473   +P cpp 10475  <P cltp 10477   ~R cer 10478  Rcnr 10479   <R cltr 10485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-ec 8393  df-qs 8397  df-ni 10486  df-pli 10487  df-mi 10488  df-lti 10489  df-plpq 10522  df-mpq 10523  df-ltpq 10524  df-enq 10525  df-nq 10526  df-erq 10527  df-plq 10528  df-mq 10529  df-1nq 10530  df-rq 10531  df-ltnq 10532  df-np 10595  df-plp 10597  df-ltp 10599  df-enr 10669  df-nr 10670  df-ltr 10673
This theorem is referenced by:  gt0srpr  10692  ltsosr  10708  0lt1sr  10709  ltasr  10714  mappsrpr  10722  ltpsrpr  10723  map2psrpr  10724
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