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Theorem submarchi 30121
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 17613 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2 eqid 2764 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
3 eqid 2764 . . . . . . 7 (0g𝑊) = (0g𝑊)
4 eqid 2764 . . . . . . 7 (.g𝑊) = (.g𝑊)
5 eqid 2764 . . . . . . 7 (le‘𝑊) = (le‘𝑊)
6 eqid 2764 . . . . . . 7 (lt‘𝑊) = (lt‘𝑊)
72, 3, 4, 5, 6isarchi2 30120 . . . . . 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 586 . . . . 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 468 . . . 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 642 . . 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 2764 . . . . . . . 8 (𝑊s 𝐴) = (𝑊s 𝐴)
1211submbas 17622 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊s 𝐴)))
132submss 17617 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
1412, 13eqsstr3d 3799 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊))
15 ssralv 3825 . . . . . . . 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 3109 . . . . . . 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 3825 . . . . . . 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 473 . . . 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 17623 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2221ad2antrr 717 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2311, 5ressle 16326 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
2423difeq1d 3888 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊s 𝐴)) ∖ I ))
255, 6pltfval 17226 . . . . . . . . . . . 12 (𝑊 ∈ Mnd → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
261, 25syl 17 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
2711submmnd 17621 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (𝑊s 𝐴) ∈ Mnd)
28 eqid 2764 . . . . . . . . . . . . 13 (le‘(𝑊s 𝐴)) = (le‘(𝑊s 𝐴))
29 eqid 2764 . . . . . . . . . . . . 13 (lt‘(𝑊s 𝐴)) = (lt‘(𝑊s 𝐴))
3028, 29pltfval 17226 . . . . . . . . . . . 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 2808 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
3332ad2antrr 717 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
34 eqidd 2765 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → 𝑥 = 𝑥)
3522, 33, 34breq123d 4822 . . . . . . . 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 2765 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
3723ad3antrrr 721 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
38 simplll 791 . . . . . . . . . . 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 11597 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41 simpllr 793 . . . . . . . . . . . 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 2846 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥𝐴)
44 eqid 2764 . . . . . . . . . . . 12 (.g‘(𝑊s 𝐴)) = (.g‘(𝑊s 𝐴))
454, 11, 44submmulg 17851 . . . . . . . . . . 11 ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0𝑥𝐴) → (𝑛(.g𝑊)𝑥) = (𝑛(.g‘(𝑊s 𝐴))𝑥))
4638, 40, 43, 45syl3anc 1490 . . . . . . . . . 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 4822 . . . . . . . . 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 3195 . . . . . . . 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 335 . . . . . . 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 3131 . . . . . 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 3131 . . . . 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 473 . . . 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 230 . . 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 30041 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Toset)
5627adantl 473 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Mnd)
57 eqid 2764 . . . . 5 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
58 eqid 2764 . . . . 5 (0g‘(𝑊s 𝐴)) = (0g‘(𝑊s 𝐴))
5957, 58, 44, 28, 29isarchi2 30120 . . . 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 579 . . 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 706 . 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 248 1 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3054  wrex 3055  cdif 3728  wss 3731   class class class wbr 4808   I cid 5183  cfv 6067  (class class class)co 6841  cn 11273  0cn0 11537  Basecbs 16131  s cress 16132  lecple 16222  0gc0g 16367  ltcplt 17208  Tosetctos 17300  Mndcmnd 17561  SubMndcsubmnd 17601  .gcmg 17808  Archicarchi 30112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-2 11334  df-3 11335  df-4 11336  df-5 11337  df-6 11338  df-7 11339  df-8 11340  df-9 11341  df-n0 11538  df-z 11624  df-dec 11740  df-uz 11886  df-fz 12533  df-seq 13008  df-ndx 16134  df-slot 16135  df-base 16137  df-sets 16138  df-ress 16139  df-plusg 16228  df-ple 16235  df-0g 16369  df-proset 17195  df-poset 17213  df-plt 17225  df-toset 17301  df-mgm 17509  df-sgrp 17551  df-mnd 17562  df-submnd 17603  df-mulg 17809  df-inftm 30113  df-archi 30114
This theorem is referenced by:  nn0archi  30224
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