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Theorem ltsrpr 9883
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 enrex 9873 . 2 ~R ∈ V
2 enrer 9871 . . 3 ~R Er (P × P)
3 erdm 7737 . . 3 ( ~R Er (P × P) → dom ~R = (P × P))
42, 3ax-mp 5 . 2 dom ~R = (P × P)
5 df-nr 9863 . 2 R = ((P × P) / ~R )
6 ltrelsr 9874 . 2 <R ⊆ (R × R)
7 ltrelpr 9805 . 2 <P ⊆ (P × P)
8 0npr 9799 . 2 ¬ ∅ ∈ P
9 dmplp 9819 . 2 dom +P = (P × P)
10 df-ltr 9866 . . 3 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
11 addclpr 9825 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 +P 𝑣) ∈ P)
1211ad2ant2lr 783 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑤 +P 𝑣) ∈ P)
13 addclpr 9825 . . . . . . 7 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
1413ad2ant2lr 783 . . . . . 6 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (𝐵 +P 𝐶) ∈ P)
1512, 14anim12ci 590 . . . . 5 ((((𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ ((𝐴P𝐵P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
1615an4s 868 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
17 enreceq 9872 . . . . . 6 (((𝑧P𝑤P) ∧ (𝐴P𝐵P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18 enreceq 9872 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19 eqcom 2627 . . . . . . 7 ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
2018, 19syl6bb 276 . . . . . 6 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷)))
2117, 20bi2anan9 916 . . . . 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 6644 . . . . . 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 9828 . . . . . . . . . 10 (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
2423oveq1i 6645 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25 addasspr 9829 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26 addasspr 9829 . . . . . . . . 9 ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
2724, 25, 263eqtr3i 2650 . . . . . . . 8 (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
2827oveq2i 6646 . . . . . . 7 (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29 addasspr 9829 . . . . . . 7 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30 addasspr 9829 . . . . . . 7 ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
3128, 29, 303eqtr4i 2652 . . . . . 6 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32 addcompr 9828 . . . . . . . . . 10 (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
3332oveq1i 6645 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34 addasspr 9829 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35 addasspr 9829 . . . . . . . . 9 ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
3633, 34, 353eqtr3i 2650 . . . . . . . 8 (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
3736oveq2i 6646 . . . . . . 7 (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38 addasspr 9829 . . . . . . 7 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39 addasspr 9829 . . . . . . 7 ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
4037, 38, 393eqtr4i 2652 . . . . . 6 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
4122, 31, 403eqtr4g 2679 . . . . 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 243 . . . 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 6663 . . . . 5 (𝑧 +P 𝑢) ∈ V
44 ovex 6663 . . . . 5 (𝐵 +P 𝐶) ∈ V
45 ltapr 9852 . . . . 5 (𝑓P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46 ovex 6663 . . . . 5 (𝑤 +P 𝑣) ∈ V
47 addcompr 9828 . . . . 5 (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48 ovex 6663 . . . . 5 (𝐴 +P 𝐷) ∈ V
4943, 44, 45, 46, 47, 48caovord3 6832 . . . 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 567 . . 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))))
511, 2, 5, 10, 50brecop 7825 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))
521, 4, 5, 6, 7, 8, 9, 51brecop2 7826 1 ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wcel 1988  cop 4174   class class class wbr 4644   × cxp 5102  dom cdm 5104  (class class class)co 6635   Er wer 7724  [cec 7725  Pcnp 9666   +P cpp 9668  <P cltp 9670   ~R cer 9671  Rcnr 9672   <R cltr 9678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-ec 7729  df-qs 7733  df-ni 9679  df-pli 9680  df-mi 9681  df-lti 9682  df-plpq 9715  df-mpq 9716  df-ltpq 9717  df-enq 9718  df-nq 9719  df-erq 9720  df-plq 9721  df-mq 9722  df-1nq 9723  df-rq 9724  df-ltnq 9725  df-np 9788  df-plp 9790  df-ltp 9792  df-enr 9862  df-nr 9863  df-ltr 9866
This theorem is referenced by:  gt0srpr  9884  ltsosr  9900  0lt1sr  9901  ltasr  9906  mappsrpr  9914  ltpsrpr  9915  map2psrpr  9916
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