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Theorem ltsrpr 11101
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 11087 . . 3 ~R Er (P × P)
2 erdm 8735 . . 3 ( ~R Er (P × P) → dom ~R = (P × P))
31, 2ax-mp 5 . 2 dom ~R = (P × P)
4 df-nr 11080 . 2 R = ((P × P) / ~R )
5 ltrelsr 11092 . 2 <R ⊆ (R × R)
6 ltrelpr 11022 . 2 <P ⊆ (P × P)
7 0npr 11016 . 2 ¬ ∅ ∈ P
8 dmplp 11036 . 2 dom +P = (P × P)
9 enrex 11091 . . 3 ~R ∈ V
10 df-ltr 11083 . . 3 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
11 addclpr 11042 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 +P 𝑣) ∈ P)
1211ad2ant2lr 747 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑤 +P 𝑣) ∈ P)
13 addclpr 11042 . . . . . . 7 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
1413ad2ant2lr 747 . . . . . 6 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (𝐵 +P 𝐶) ∈ P)
1512, 14anim12ci 613 . . . . 5 ((((𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ ((𝐴P𝐵P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
1615an4s 659 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
17 enreceq 11090 . . . . . 6 (((𝑧P𝑤P) ∧ (𝐴P𝐵P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18 enreceq 11090 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19 eqcom 2735 . . . . . . 7 ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
2018, 19bitrdi 287 . . . . . 6 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷)))
2117, 20bi2anan9 637 . . . . 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 7429 . . . . . 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 11045 . . . . . . . . . 10 (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
2423oveq1i 7430 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25 addasspr 11046 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26 addasspr 11046 . . . . . . . . 9 ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
2724, 25, 263eqtr3i 2764 . . . . . . . 8 (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
2827oveq2i 7431 . . . . . . 7 (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29 addasspr 11046 . . . . . . 7 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30 addasspr 11046 . . . . . . 7 ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
3128, 29, 303eqtr4i 2766 . . . . . 6 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32 addcompr 11045 . . . . . . . . . 10 (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
3332oveq1i 7430 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34 addasspr 11046 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35 addasspr 11046 . . . . . . . . 9 ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
3633, 34, 353eqtr3i 2764 . . . . . . . 8 (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
3736oveq2i 7431 . . . . . . 7 (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38 addasspr 11046 . . . . . . 7 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39 addasspr 11046 . . . . . . 7 ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
4037, 38, 393eqtr4i 2766 . . . . . 6 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
4122, 31, 403eqtr4g 2793 . . . . 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, 41biimtrdi 252 . . . 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 7453 . . . . 5 (𝑧 +P 𝑢) ∈ V
44 ovex 7453 . . . . 5 (𝐵 +P 𝐶) ∈ V
45 ltapr 11069 . . . . 5 (𝑓P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46 ovex 7453 . . . . 5 (𝑤 +P 𝑣) ∈ V
47 addcompr 11045 . . . . 5 (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48 ovex 7453 . . . . 5 (𝐴 +P 𝐷) ∈ V
4943, 44, 45, 46, 47, 48caovord3 7634 . . . 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 683 . . 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 8829 . 2 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))
523, 4, 5, 6, 7, 8, 51brecop2 8830 1 ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  cop 4635   class class class wbr 5148   × cxp 5676  dom cdm 5678  (class class class)co 7420   Er wer 8722  [cec 8723  Pcnp 10883   +P cpp 10885  <P cltp 10887   ~R cer 10888  Rcnr 10889   <R cltr 10895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-ec 8727  df-qs 8731  df-ni 10896  df-pli 10897  df-mi 10898  df-lti 10899  df-plpq 10932  df-mpq 10933  df-ltpq 10934  df-enq 10935  df-nq 10936  df-erq 10937  df-plq 10938  df-mq 10939  df-1nq 10940  df-rq 10941  df-ltnq 10942  df-np 11005  df-plp 11007  df-ltp 11009  df-enr 11079  df-nr 11080  df-ltr 11083
This theorem is referenced by:  gt0srpr  11102  ltsosr  11118  0lt1sr  11119  ltasr  11124  mappsrpr  11132  ltpsrpr  11133  map2psrpr  11134
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