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Theorem ltsrprg 7548
 Description: Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
Assertion
Ref Expression
ltsrprg (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))

Proof of Theorem ltsrprg
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrex 7538 . 2 ~R ∈ V
2 enrer 7536 . 2 ~R Er (P × P)
3 df-nr 7528 . 2 R = ((P × P) / ~R )
4 df-ltr 7531 . 2 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
5 enreceq 7537 . . . . 5 (((𝑧P𝑤P) ∧ (𝐴P𝐵P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
6 enreceq 7537 . . . . . 6 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
7 eqcom 2139 . . . . . 6 ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
86, 7syl6bb 195 . . . . 5 (((𝑣P𝑢P) ∧ (𝐶P𝐷P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷)))
95, 8bi2anan9 595 . . . 4 ((((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))))
10 oveq12 5776 . . . . . . 7 (((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)))
1110adantl 275 . . . . . 6 (((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) ∧ ((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)))
12 simprlr 527 . . . . . . . . . . 11 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝑢P)
13 simplrr 525 . . . . . . . . . . 11 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝐵P)
14 addcomprg 7379 . . . . . . . . . . . 12 ((𝑢P𝐵P) → (𝑢 +P 𝐵) = (𝐵 +P 𝑢))
1514oveq1d 5782 . . . . . . . . . . 11 ((𝑢P𝐵P) → ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶))
1612, 13, 15syl2anc 408 . . . . . . . . . 10 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶))
17 simprrl 528 . . . . . . . . . . 11 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝐶P)
18 addassprg 7380 . . . . . . . . . . 11 ((𝑢P𝐵P𝐶P) → ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶)))
1912, 13, 17, 18syl3anc 1216 . . . . . . . . . 10 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶)))
20 addassprg 7380 . . . . . . . . . . 11 ((𝐵P𝑢P𝐶P) → ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶)))
2113, 12, 17, 20syl3anc 1216 . . . . . . . . . 10 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶)))
2216, 19, 213eqtr3d 2178 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶)))
2322oveq2d 5783 . . . . . . . 8 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶))))
24 simplll 522 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝑧P)
25 addclpr 7338 . . . . . . . . . . . . 13 ((𝑤P𝑣P) → (𝑤 +P 𝑣) ∈ P)
2625ad2ant2lr 501 . . . . . . . . . . . 12 (((𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (𝑤 +P 𝑣) ∈ P)
27 addclpr 7338 . . . . . . . . . . . . 13 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2827ad2ant2lr 501 . . . . . . . . . . . 12 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (𝐵 +P 𝐶) ∈ P)
2926, 28anim12ci 337 . . . . . . . . . . 11 ((((𝑧P𝑤P) ∧ (𝑣P𝑢P)) ∧ ((𝐴P𝐵P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
3029an4s 577 . . . . . . . . . 10 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
3130simpld 111 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝐵 +P 𝐶) ∈ P)
32 addassprg 7380 . . . . . . . . 9 ((𝑧P𝑢P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))))
3324, 12, 31, 32syl3anc 1216 . . . . . . . 8 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))))
34 addclpr 7338 . . . . . . . . . 10 ((𝑢P𝐶P) → (𝑢 +P 𝐶) ∈ P)
3512, 17, 34syl2anc 408 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑢 +P 𝐶) ∈ P)
36 addassprg 7380 . . . . . . . . 9 ((𝑧P𝐵P ∧ (𝑢 +P 𝐶) ∈ P) → ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶))))
3724, 13, 35, 36syl3anc 1216 . . . . . . . 8 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶))))
3823, 33, 373eqtr4d 2180 . . . . . . 7 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)))
3938adantr 274 . . . . . 6 (((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) ∧ ((𝑧 +P 𝐵) = (𝑤 +P 𝐴) ∧ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))) → ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)))
40 simprll 526 . . . . . . . . . . . 12 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝑣P)
41 simplrl 524 . . . . . . . . . . . 12 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝐴P)
42 addcomprg 7379 . . . . . . . . . . . 12 ((𝑣P𝐴P) → (𝑣 +P 𝐴) = (𝐴 +P 𝑣))
4340, 41, 42syl2anc 408 . . . . . . . . . . 11 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑣 +P 𝐴) = (𝐴 +P 𝑣))
4443oveq1d 5782 . . . . . . . . . 10 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷))
45 simprrr 529 . . . . . . . . . . 11 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝐷P)
46 addassprg 7380 . . . . . . . . . . 11 ((𝑣P𝐴P𝐷P) → ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷)))
4740, 41, 45, 46syl3anc 1216 . . . . . . . . . 10 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷)))
48 addassprg 7380 . . . . . . . . . . 11 ((𝐴P𝑣P𝐷P) → ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷)))
4941, 40, 45, 48syl3anc 1216 . . . . . . . . . 10 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷)))
5044, 47, 493eqtr3d 2178 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷)))
5150oveq2d 5783 . . . . . . . 8 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷))))
52 simpllr 523 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → 𝑤P)
53 addclpr 7338 . . . . . . . . . 10 ((𝐴P𝐷P) → (𝐴 +P 𝐷) ∈ P)
5441, 45, 53syl2anc 408 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝐴 +P 𝐷) ∈ P)
55 addassprg 7380 . . . . . . . . 9 ((𝑤P𝑣P ∧ (𝐴 +P 𝐷) ∈ P) → ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))))
5652, 40, 54, 55syl3anc 1216 . . . . . . . 8 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))))
57 addclpr 7338 . . . . . . . . . 10 ((𝑣P𝐷P) → (𝑣 +P 𝐷) ∈ P)
5840, 45, 57syl2anc 408 . . . . . . . . 9 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑣 +P 𝐷) ∈ P)
59 addassprg 7380 . . . . . . . . 9 ((𝑤P𝐴P ∧ (𝑣 +P 𝐷) ∈ P) → ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷))))
6052, 41, 58, 59syl3anc 1216 . . . . . . . 8 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷))))
6151, 56, 603eqtr4d 2180 . . . . . . 7 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)))
6261adantr 274 . . . . . 6 (((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) ∧ ((𝑧 +P 𝐵) = (𝑤 +P 𝐴) ∧ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))) → ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)))
6311, 39, 623eqtr4d 2180 . . . . 5 (((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) ∧ ((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)))
6463ex 114 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (((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))))
659, 64sylbid 149 . . 3 ((((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))))
66 ltaprg 7420 . . . . 5 ((𝑥P𝑦P𝑓P) → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
6766adantl 275 . . . 4 (((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) ∧ (𝑥P𝑦P𝑓P)) → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
68 addclpr 7338 . . . . 5 ((𝑧P𝑢P) → (𝑧 +P 𝑢) ∈ P)
6924, 12, 68syl2anc 408 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑧 +P 𝑢) ∈ P)
7030simprd 113 . . . 4 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (𝑤 +P 𝑣) ∈ P)
71 addcomprg 7379 . . . . 5 ((𝑥P𝑦P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
7271adantl 275 . . . 4 (((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) ∧ (𝑥P𝑦P)) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
7367, 69, 31, 70, 72, 54caovord3d 5934 . . 3 ((((𝑧P𝑤P) ∧ (𝐴P𝐵P)) ∧ ((𝑣P𝑢P) ∧ (𝐶P𝐷P))) → (((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))))
7465, 73syld 45 . 2 ((((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))))
751, 2, 3, 4, 74brecop 6512 1 (((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ⟨cop 3525   class class class wbr 3924  (class class class)co 5767  [cec 6420  Pcnp 7092   +P cpp 7094
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