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 29522
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 17267 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2 eqid 2621 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
3 eqid 2621 . . . . . . 7 (0g𝑊) = (0g𝑊)
4 eqid 2621 . . . . . . 7 (.g𝑊) = (.g𝑊)
5 eqid 2621 . . . . . . 7 (le‘𝑊) = (le‘𝑊)
6 eqid 2621 . . . . . . 7 (lt‘𝑊) = (lt‘𝑊)
72, 3, 4, 5, 6isarchi2 29521 . . . . . 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 491 . . . . 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 501 . . . 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 845 . . 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 2621 . . . . . . . 8 (𝑊s 𝐴) = (𝑊s 𝐴)
1211submbas 17276 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊s 𝐴)))
132submss 17271 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
1412, 13eqsstr3d 3619 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊))
15 ssralv 3645 . . . . . . . 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 2957 . . . . . . 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 3645 . . . . . . 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 47 . . . . . 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 482 . . . 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 17277 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2221ad2antrr 761 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2311, 5ressle 15980 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
2423difeq1d 3705 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊s 𝐴)) ∖ I ))
255, 6pltfval 16880 . . . . . . . . . . . 12 (𝑊 ∈ Mnd → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
261, 25syl 17 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
2711submmnd 17275 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (𝑊s 𝐴) ∈ Mnd)
28 eqid 2621 . . . . . . . . . . . . 13 (le‘(𝑊s 𝐴)) = (le‘(𝑊s 𝐴))
29 eqid 2621 . . . . . . . . . . . . 13 (lt‘(𝑊s 𝐴)) = (lt‘(𝑊s 𝐴))
3028, 29pltfval 16880 . . . . . . . . . . . 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 2665 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
3332ad2antrr 761 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
34 eqidd 2622 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → 𝑥 = 𝑥)
3522, 33, 34breq123d 4627 . . . . . . . 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 2622 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
3723ad3antrrr 765 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
38 simplll 797 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (SubMnd‘𝑊))
39 simpr 477 . . . . . . . . . . . 12 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
4039nnnn0d 11295 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41 simpllr 798 . . . . . . . . . . . 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 2701 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥𝐴)
44 eqid 2621 . . . . . . . . . . . 12 (.g‘(𝑊s 𝐴)) = (.g‘(𝑊s 𝐴))
454, 11, 44submmulg 17507 . . . . . . . . . . 11 ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0𝑥𝐴) → (𝑛(.g𝑊)𝑥) = (𝑛(.g‘(𝑊s 𝐴))𝑥))
4638, 40, 43, 45syl3anc 1323 . . . . . . . . . 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 4627 . . . . . . . . 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 3042 . . . . . . . 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 334 . . . . . . 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 2979 . . . . . 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 2979 . . . . 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 482 . . . 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 229 . . 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 29442 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Toset)
5627adantl 482 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Mnd)
57 eqid 2621 . . . . 5 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
58 eqid 2621 . . . . 5 (0g‘(𝑊s 𝐴)) = (0g‘(𝑊s 𝐴))
5957, 58, 44, 28, 29isarchi2 29521 . . . 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 692 . . 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 750 . 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 247 1 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cdif 3552  wss 3555   class class class wbr 4613   I cid 4984  cfv 5847  (class class class)co 6604  cn 10964  0cn0 11236  Basecbs 15781  s cress 15782  lecple 15869  0gc0g 16021  ltcplt 16862  Tosetctos 16954  Mndcmnd 17215  SubMndcsubmnd 17255  .gcmg 17461  Archicarchi 29513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-seq 12742  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-ple 15882  df-0g 16023  df-preset 16849  df-poset 16867  df-plt 16879  df-toset 16955  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-inftm 29514  df-archi 29515
This theorem is referenced by:  nn0archi  29625
  Copyright terms: Public domain W3C validator