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Theorem ltsrpr 9777
 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 9767 . 2 ~R ∈ V
2 enrer 9765 . . 3 ~R Er (P × P)
3 erdm 7639 . . 3 ( ~R Er (P × P) → dom ~R = (P × P))
42, 3ax-mp 5 . 2 dom ~R = (P × P)
5 df-nr 9757 . 2 R = ((P × P) / ~R )
6 ltrelsr 9768 . 2 <R ⊆ (R × R)
7 ltrelpr 9699 . 2 <P ⊆ (P × P)
8 0npr 9693 . 2 ¬ ∅ ∈ P
9 dmplp 9713 . 2 dom +P = (P × P)
10 df-ltr 9760 . . 3 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
11 addclpr 9719 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 +P 𝑣) ∈ P)
1211ad2ant2lr 780 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑤 +P 𝑣) ∈ P)
13 addclpr 9719 . . . . . . 7 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
1413ad2ant2lr 780 . . . . . 6 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (𝐵 +P 𝐶) ∈ P)
1512, 14anim12ci 589 . . . . 5 ((((𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ ((𝐴P𝐵P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
1615an4s 865 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
17 enreceq 9766 . . . . . 6 (((𝑧P𝑤P) ∧ (𝐴P𝐵P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18 enreceq 9766 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19 eqcom 2617 . . . . . . 7 ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
2018, 19syl6bb 275 . . . . . 6 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷)))
2117, 20bi2anan9 913 . . . . 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 6558 . . . . . 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 9722 . . . . . . . . . 10 (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
2423oveq1i 6559 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25 addasspr 9723 . . . . . . . . 9 ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26 addasspr 9723 . . . . . . . . 9 ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
2724, 25, 263eqtr3i 2640 . . . . . . . 8 (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
2827oveq2i 6560 . . . . . . 7 (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29 addasspr 9723 . . . . . . 7 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30 addasspr 9723 . . . . . . 7 ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
3128, 29, 303eqtr4i 2642 . . . . . 6 ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32 addcompr 9722 . . . . . . . . . 10 (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
3332oveq1i 6559 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34 addasspr 9723 . . . . . . . . 9 ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35 addasspr 9723 . . . . . . . . 9 ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
3633, 34, 353eqtr3i 2640 . . . . . . . 8 (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
3736oveq2i 6560 . . . . . . 7 (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38 addasspr 9723 . . . . . . 7 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39 addasspr 9723 . . . . . . 7 ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
4037, 38, 393eqtr4i 2642 . . . . . 6 ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
4122, 31, 403eqtr4g 2669 . . . . 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 242 . . . 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 6577 . . . . 5 (𝑧 +P 𝑢) ∈ V
44 ovex 6577 . . . . 5 (𝐵 +P 𝐶) ∈ V
45 ltapr 9746 . . . . 5 (𝑓P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46 ovex 6577 . . . . 5 (𝑤 +P 𝑣) ∈ V
47 addcompr 9722 . . . . 5 (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48 ovex 6577 . . . . 5 (𝐴 +P 𝐷) ∈ V
4943, 44, 45, 46, 47, 48caovord3 6745 . . . 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 566 . . 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 7727 . 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 7728 1 ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  (class class class)co 6549   Er wer 7626  [cec 7627  Pcnp 9560   +P cpp 9562
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