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Theorem ltsosr 7763
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 7762 . 2 <R Po R
2 df-nr 7726 . . . 4 R = ((P × P) / ~R )
3 breq1 4007 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R [⟨𝑐, 𝑑⟩] ~R ))
4 breq1 4007 . . . . . 6 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R [⟨𝑒, 𝑓⟩] ~R ))
54orbi1d 791 . . . . 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 234 . . . 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 4008 . . . . 5 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → (𝑥 <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R 𝑦))
8 breq2 4008 . . . . . 6 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))
98orbi2d 790 . . . . 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 234 . . . 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 4008 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R 𝑧))
12 breq1 4007 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ([⟨𝑒, 𝑓⟩] ~R <R 𝑦𝑧 <R 𝑦))
1311, 12orbi12d 793 . . . . 5 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦) ↔ (𝑥 <R 𝑧𝑧 <R 𝑦)))
1413imbi2d 230 . . . 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 1021 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑎P)
16 simp3r 1026 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑓P)
17 addclpr 7536 . . . . . . . . 9 ((𝑎P𝑓P) → (𝑎 +P 𝑓) ∈ P)
1815, 16, 17syl2anc 411 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑓) ∈ P)
19 simp2r 1024 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑑P)
20 addclpr 7536 . . . . . . . 8 (((𝑎 +P 𝑓) ∈ P𝑑P) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
2118, 19, 20syl2anc 411 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
22 simp2l 1023 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑐P)
23 addclpr 7536 . . . . . . . . 9 ((𝑓P𝑐P) → (𝑓 +P 𝑐) ∈ P)
2416, 22, 23syl2anc 411 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P 𝑐) ∈ P)
25 simp1r 1022 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑏P)
26 addclpr 7536 . . . . . . . 8 (((𝑓 +P 𝑐) ∈ P𝑏P) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
2724, 25, 26syl2anc 411 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
28 simp3l 1025 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑒P)
29 addclpr 7536 . . . . . . . . 9 ((𝑏P𝑒P) → (𝑏 +P 𝑒) ∈ P)
3025, 28, 29syl2anc 411 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑒) ∈ P)
31 addclpr 7536 . . . . . . . 8 (((𝑏 +P 𝑒) ∈ P𝑑P) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
3230, 19, 31syl2anc 411 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
33 ltsopr 7595 . . . . . . . 8 <P Or P
34 sowlin 4321 . . . . . . . 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 424 . . . . . . 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 1238 . . . . . 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 7536 . . . . . . . . 9 ((𝑎P𝑑P) → (𝑎 +P 𝑑) ∈ P)
3815, 19, 37syl2anc 411 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑑) ∈ P)
39 addclpr 7536 . . . . . . . . 9 ((𝑏P𝑐P) → (𝑏 +P 𝑐) ∈ P)
4025, 22, 39syl2anc 411 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑐) ∈ P)
41 ltaprg 7618 . . . . . . . 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 1238 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
43 addcomprg 7577 . . . . . . . . . . 11 ((𝑟P𝑠P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
4443adantl 277 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
45 addassprg 7578 . . . . . . . . . . 11 ((𝑟P𝑠P𝑡P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4645adantl 277 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4716, 15, 19, 44, 46caov12d 6056 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = (𝑎 +P (𝑓 +P 𝑑)))
4846, 15, 16, 19caovassd 6034 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) = (𝑎 +P (𝑓 +P 𝑑)))
4947, 48eqtr4d 2213 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = ((𝑎 +P 𝑓) +P 𝑑))
5046, 16, 25, 22caovassd 6034 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = (𝑓 +P (𝑏 +P 𝑐)))
5116, 25, 22, 44, 46caov32d 6055 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = ((𝑓 +P 𝑐) +P 𝑏))
5250, 51eqtr3d 2212 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑏 +P 𝑐)) = ((𝑓 +P 𝑐) +P 𝑏))
5349, 52breq12d 4017 . . . . . . 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 188 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
55 ltaprg 7618 . . . . . . . . 9 ((𝑟P𝑠P𝑡P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5655adantl 277 . . . . . . . 8 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5756, 18, 30, 19, 44caovord2d 6044 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑)))
58 addclpr 7536 . . . . . . . . . 10 ((𝑒P𝑑P) → (𝑒 +P 𝑑) ∈ P)
5928, 19, 58syl2anc 411 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑒 +P 𝑑) ∈ P)
6056, 59, 24, 25, 44caovord2d 6044 . . . . . . . 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 6034 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = (𝑏 +P (𝑒 +P 𝑑)))
6244, 25, 59caovcomd 6031 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P (𝑒 +P 𝑑)) = ((𝑒 +P 𝑑) +P 𝑏))
6361, 62eqtrd 2210 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = ((𝑒 +P 𝑑) +P 𝑏))
6463breq1d 4014 . . . . . . . 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 191 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
6657, 65orbi12d 793 . . . . . 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 203 . . . . 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 7746 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
69683adant3 1017 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
70 ltsrprg 7746 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
71703adant2 1016 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
72 ltsrprg 7746 . . . . . . . 8 (((𝑒P𝑓P) ∧ (𝑐P𝑑P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7372ancoms 268 . . . . . . 7 (((𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
74733adant1 1015 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7571, 74orbi12d 793 . . . . 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 203 . . . 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 6624 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
7877rgen3 2564 . 2 𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))
79 df-iso 4298 . 2 ( <R Or R ↔ ( <R Po R ∧ ∀𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))))
801, 78, 79mpbir2an 942 1 <R Or R
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2148  wral 2455  cop 3596   class class class wbr 4004   Po wpo 4295   Or wor 4296  (class class class)co 5875  [cec 6533  Pcnp 7290   +P cpp 7292  <P cltp 7294   ~R cer 7295  Rcnr 7296   <R cltr 7302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-iltp 7469  df-enr 7725  df-nr 7726  df-ltr 7729
This theorem is referenced by:  1ne0sr  7765  addgt0sr  7774  caucvgsrlemcl  7788  caucvgsrlemfv  7790  suplocsrlemb  7805  suplocsrlempr  7806  suplocsrlem  7807  axpre-ltirr  7881  axpre-ltwlin  7882  axpre-lttrn  7883
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