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Theorem ltsosr 7586
 Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
Assertion
Ref Expression
ltsosr <R Or R

Proof of Theorem ltsosr
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltposr 7585 . 2 <R Po R
2 df-nr 7549 . . . 4 R = ((P × P) / ~R )
3 breq1 3932 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R [⟨𝑐, 𝑑⟩] ~R ))
4 breq1 3932 . . . . . 6 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R [⟨𝑒, 𝑓⟩] ~R ))
54orbi1d 780 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )))
63, 5imbi12d 233 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )) ↔ (𝑥 <R [⟨𝑐, 𝑑⟩] ~R → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ))))
7 breq2 3933 . . . . 5 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → (𝑥 <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R 𝑦))
8 breq2 3933 . . . . . 6 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))
98orbi2d 779 . . . . 5 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦)))
107, 9imbi12d 233 . . . 4 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ((𝑥 <R [⟨𝑐, 𝑑⟩] ~R → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )) ↔ (𝑥 <R 𝑦 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))))
11 breq2 3933 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R 𝑧))
12 breq1 3932 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ([⟨𝑒, 𝑓⟩] ~R <R 𝑦𝑧 <R 𝑦))
1311, 12orbi12d 782 . . . . 5 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦) ↔ (𝑥 <R 𝑧𝑧 <R 𝑦)))
1413imbi2d 229 . . . 4 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R 𝑦 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦)) ↔ (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))))
15 simp1l 1005 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑎P)
16 simp3r 1010 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑓P)
17 addclpr 7359 . . . . . . . . 9 ((𝑎P𝑓P) → (𝑎 +P 𝑓) ∈ P)
1815, 16, 17syl2anc 408 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑓) ∈ P)
19 simp2r 1008 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑑P)
20 addclpr 7359 . . . . . . . 8 (((𝑎 +P 𝑓) ∈ P𝑑P) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
2118, 19, 20syl2anc 408 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
22 simp2l 1007 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑐P)
23 addclpr 7359 . . . . . . . . 9 ((𝑓P𝑐P) → (𝑓 +P 𝑐) ∈ P)
2416, 22, 23syl2anc 408 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P 𝑐) ∈ P)
25 simp1r 1006 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑏P)
26 addclpr 7359 . . . . . . . 8 (((𝑓 +P 𝑐) ∈ P𝑏P) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
2724, 25, 26syl2anc 408 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
28 simp3l 1009 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑒P)
29 addclpr 7359 . . . . . . . . 9 ((𝑏P𝑒P) → (𝑏 +P 𝑒) ∈ P)
3025, 28, 29syl2anc 408 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑒) ∈ P)
31 addclpr 7359 . . . . . . . 8 (((𝑏 +P 𝑒) ∈ P𝑑P) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
3230, 19, 31syl2anc 408 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
33 ltsopr 7418 . . . . . . . 8 <P Or P
34 sowlin 4242 . . . . . . . 8 ((<P Or P ∧ (((𝑎 +P 𝑓) +P 𝑑) ∈ P ∧ ((𝑓 +P 𝑐) +P 𝑏) ∈ P ∧ ((𝑏 +P 𝑒) +P 𝑑) ∈ P)) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
3533, 34mpan 420 . . . . . . 7 ((((𝑎 +P 𝑓) +P 𝑑) ∈ P ∧ ((𝑓 +P 𝑐) +P 𝑏) ∈ P ∧ ((𝑏 +P 𝑒) +P 𝑑) ∈ P) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
3621, 27, 32, 35syl3anc 1216 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
37 addclpr 7359 . . . . . . . . 9 ((𝑎P𝑑P) → (𝑎 +P 𝑑) ∈ P)
3815, 19, 37syl2anc 408 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑑) ∈ P)
39 addclpr 7359 . . . . . . . . 9 ((𝑏P𝑐P) → (𝑏 +P 𝑐) ∈ P)
4025, 22, 39syl2anc 408 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑐) ∈ P)
41 ltaprg 7441 . . . . . . . 8 (((𝑎 +P 𝑑) ∈ P ∧ (𝑏 +P 𝑐) ∈ P𝑓P) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
4238, 40, 16, 41syl3anc 1216 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
43 addcomprg 7400 . . . . . . . . . . 11 ((𝑟P𝑠P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
4443adantl 275 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
45 addassprg 7401 . . . . . . . . . . 11 ((𝑟P𝑠P𝑡P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4645adantl 275 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4716, 15, 19, 44, 46caov12d 5952 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = (𝑎 +P (𝑓 +P 𝑑)))
4846, 15, 16, 19caovassd 5930 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) = (𝑎 +P (𝑓 +P 𝑑)))
4947, 48eqtr4d 2175 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = ((𝑎 +P 𝑓) +P 𝑑))
5046, 16, 25, 22caovassd 5930 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = (𝑓 +P (𝑏 +P 𝑐)))
5116, 25, 22, 44, 46caov32d 5951 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = ((𝑓 +P 𝑐) +P 𝑏))
5250, 51eqtr3d 2174 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑏 +P 𝑐)) = ((𝑓 +P 𝑐) +P 𝑏))
5349, 52breq12d 3942 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐)) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
5442, 53bitrd 187 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
55 ltaprg 7441 . . . . . . . . 9 ((𝑟P𝑠P𝑡P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5655adantl 275 . . . . . . . 8 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5756, 18, 30, 19, 44caovord2d 5940 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑)))
58 addclpr 7359 . . . . . . . . . 10 ((𝑒P𝑑P) → (𝑒 +P 𝑑) ∈ P)
5928, 19, 58syl2anc 408 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑒 +P 𝑑) ∈ P)
6056, 59, 24, 25, 44caovord2d 5940 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑒 +P 𝑑) +P 𝑏)<P ((𝑓 +P 𝑐) +P 𝑏)))
6146, 25, 28, 19caovassd 5930 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = (𝑏 +P (𝑒 +P 𝑑)))
6244, 25, 59caovcomd 5927 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P (𝑒 +P 𝑑)) = ((𝑒 +P 𝑑) +P 𝑏))
6361, 62eqtrd 2172 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = ((𝑒 +P 𝑑) +P 𝑏))
6463breq1d 3939 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) ↔ ((𝑒 +P 𝑑) +P 𝑏)<P ((𝑓 +P 𝑐) +P 𝑏)))
6560, 64bitr4d 190 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
6657, 65orbi12d 782 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)) ↔ (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
6736, 54, 663imtr4d 202 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐))))
68 ltsrprg 7569 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
69683adant3 1001 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
70 ltsrprg 7569 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
71703adant2 1000 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
72 ltsrprg 7569 . . . . . . . 8 (((𝑒P𝑓P) ∧ (𝑐P𝑑P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7372ancoms 266 . . . . . . 7 (((𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
74733adant1 999 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7571, 74orbi12d 782 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐))))
7667, 69, 753imtr4d 202 . . . 4 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )))
772, 6, 10, 14, 763ecoptocl 6518 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
7877rgen3 2519 . 2 𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))
79 df-iso 4219 . 2 ( <R Or R ↔ ( <R Po R ∧ ∀𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))))
801, 78, 79mpbir2an 926 1 <R Or R
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ⟨cop 3530   class class class wbr 3929   Po wpo 4216   Or wor 4217  (class class class)co 5774  [cec 6427  Pcnp 7113   +P cpp 7115
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