Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  submarchi Structured version   Visualization version   GIF version

Theorem submarchi 29071
Description: A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
Assertion
Ref Expression
submarchi (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)

Proof of Theorem submarchi
Dummy variables 𝑥 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submrcl 17169 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2 eqid 2610 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
3 eqid 2610 . . . . . . 7 (0g𝑊) = (0g𝑊)
4 eqid 2610 . . . . . . 7 (.g𝑊) = (.g𝑊)
5 eqid 2610 . . . . . . 7 (le‘𝑊) = (le‘𝑊)
6 eqid 2610 . . . . . . 7 (lt‘𝑊) = (lt‘𝑊)
72, 3, 4, 5, 6isarchi2 29070 . . . . . 6 ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
81, 7sylan2 490 . . . . 5 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
98biimpa 500 . . . 4 (((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) ∧ 𝑊 ∈ Archi) → ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)))
109an32s 842 . . 3 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)))
11 eqid 2610 . . . . . . . 8 (𝑊s 𝐴) = (𝑊s 𝐴)
1211submbas 17178 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊s 𝐴)))
132submss 17173 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
1412, 13eqsstr3d 3603 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊))
15 ssralv 3629 . . . . . . . 8 ((Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) → ∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
1615ralimdv 2946 . . . . . . 7 ((Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) → ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
17 ssralv 3629 . . . . . . 7 ((Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
1816, 17syld 46 . . . . . 6 ((Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
1914, 18syl 17 . . . . 5 (𝐴 ∈ (SubMnd‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
2019adantl 481 . . . 4 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥))))
2111, 3subm0 17179 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2221ad2antrr 758 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2311, 5ressle 15882 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
2423difeq1d 3689 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊s 𝐴)) ∖ I ))
255, 6pltfval 16782 . . . . . . . . . . . 12 (𝑊 ∈ Mnd → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
261, 25syl 17 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
2711submmnd 17177 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (𝑊s 𝐴) ∈ Mnd)
28 eqid 2610 . . . . . . . . . . . . 13 (le‘(𝑊s 𝐴)) = (le‘(𝑊s 𝐴))
29 eqid 2610 . . . . . . . . . . . . 13 (lt‘(𝑊s 𝐴)) = (lt‘(𝑊s 𝐴))
3028, 29pltfval 16782 . . . . . . . . . . . 12 ((𝑊s 𝐴) ∈ Mnd → (lt‘(𝑊s 𝐴)) = ((le‘(𝑊s 𝐴)) ∖ I ))
3127, 30syl 17 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘(𝑊s 𝐴)) = ((le‘(𝑊s 𝐴)) ∖ I ))
3224, 26, 313eqtr4d 2654 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
3332ad2antrr 758 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
34 eqidd 2611 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → 𝑥 = 𝑥)
3522, 33, 34breq123d 4597 . . . . . . . 8 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → ((0g𝑊)(lt‘𝑊)𝑥 ↔ (0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥))
36 eqidd 2611 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
3723ad3antrrr 762 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
38 simplll 794 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (SubMnd‘𝑊))
39 simpr 476 . . . . . . . . . . . 12 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
4039nnnn0d 11228 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41 simpllr 795 . . . . . . . . . . . 12 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (Base‘(𝑊s 𝐴)))
4238, 12syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 = (Base‘(𝑊s 𝐴)))
4341, 42eleqtrrd 2691 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥𝐴)
44 eqid 2610 . . . . . . . . . . . 12 (.g‘(𝑊s 𝐴)) = (.g‘(𝑊s 𝐴))
454, 11, 44submmulg 17409 . . . . . . . . . . 11 ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0𝑥𝐴) → (𝑛(.g𝑊)𝑥) = (𝑛(.g‘(𝑊s 𝐴))𝑥))
4638, 40, 43, 45syl3anc 1318 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑛(.g𝑊)𝑥) = (𝑛(.g‘(𝑊s 𝐴))𝑥))
4736, 37, 46breq123d 4597 . . . . . . . . 9 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥) ↔ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥)))
4847rexbidva 3031 . . . . . . . 8 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥)))
4935, 48imbi12d 333 . . . . . . 7 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) ↔ ((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
5049ralbidva 2968 . . . . . 6 ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) → (∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) ↔ ∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
5150ralbidva 2968 . . . . 5 (𝐴 ∈ (SubMnd‘𝑊) → (∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) ↔ ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
5251adantl 481 . . . 4 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) ↔ ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
5320, 52sylibd 228 . . 3 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘𝑊)(𝑛(.g𝑊)𝑥)) → ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
5410, 53mpd 15 . 2 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥)))
55 resstos 28991 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Toset)
5627adantl 481 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Mnd)
57 eqid 2610 . . . . 5 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
58 eqid 2610 . . . . 5 (0g‘(𝑊s 𝐴)) = (0g‘(𝑊s 𝐴))
5957, 58, 44, 28, 29isarchi2 29070 . . . 4 (((𝑊s 𝐴) ∈ Toset ∧ (𝑊s 𝐴) ∈ Mnd) → ((𝑊s 𝐴) ∈ Archi ↔ ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
6055, 56, 59syl2anc 691 . . 3 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ((𝑊s 𝐴) ∈ Archi ↔ ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
6160adantlr 747 . 2 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → ((𝑊s 𝐴) ∈ Archi ↔ ∀𝑥 ∈ (Base‘(𝑊s 𝐴))∀𝑦 ∈ (Base‘(𝑊s 𝐴))((0g‘(𝑊s 𝐴))(lt‘(𝑊s 𝐴))𝑥 → ∃𝑛 ∈ ℕ 𝑦(le‘(𝑊s 𝐴))(𝑛(.g‘(𝑊s 𝐴))𝑥))))
6254, 61mpbird 246 1 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cdif 3537  wss 3540   class class class wbr 4583   I cid 4948  cfv 5804  (class class class)co 6549  cn 10897  0cn0 11169  Basecbs 15695  s cress 15696  lecple 15775  0gc0g 15923  ltcplt 16764  Tosetctos 16856  Mndcmnd 17117  SubMndcsubmnd 17157  .gcmg 17363  Archicarchi 29062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-seq 12664  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-ple 15788  df-0g 15925  df-preset 16751  df-poset 16769  df-plt 16781  df-toset 16857  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-inftm 29063  df-archi 29064
This theorem is referenced by:  nn0archi  29174
  Copyright terms: Public domain W3C validator