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Theorem submarchi 30049
 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 17547 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2 eqid 2760 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
3 eqid 2760 . . . . . . 7 (0g𝑊) = (0g𝑊)
4 eqid 2760 . . . . . . 7 (.g𝑊) = (.g𝑊)
5 eqid 2760 . . . . . . 7 (le‘𝑊) = (le‘𝑊)
6 eqid 2760 . . . . . . 7 (lt‘𝑊) = (lt‘𝑊)
72, 3, 4, 5, 6isarchi2 30048 . . . . . 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 492 . . . . 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 502 . . . 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 881 . . 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 2760 . . . . . . . 8 (𝑊s 𝐴) = (𝑊s 𝐴)
1211submbas 17556 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊s 𝐴)))
132submss 17551 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
1412, 13eqsstr3d 3781 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊))
15 ssralv 3807 . . . . . . . 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 3101 . . . . . . 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 3807 . . . . . . 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 17557 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2221ad2antrr 764 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2311, 5ressle 16261 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
2423difeq1d 3870 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊s 𝐴)) ∖ I ))
255, 6pltfval 17160 . . . . . . . . . . . 12 (𝑊 ∈ Mnd → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
261, 25syl 17 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
2711submmnd 17555 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (𝑊s 𝐴) ∈ Mnd)
28 eqid 2760 . . . . . . . . . . . . 13 (le‘(𝑊s 𝐴)) = (le‘(𝑊s 𝐴))
29 eqid 2760 . . . . . . . . . . . . 13 (lt‘(𝑊s 𝐴)) = (lt‘(𝑊s 𝐴))
3028, 29pltfval 17160 . . . . . . . . . . . 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 2804 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
3332ad2antrr 764 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
34 eqidd 2761 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → 𝑥 = 𝑥)
3522, 33, 34breq123d 4818 . . . . . . . 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 2761 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
3723ad3antrrr 768 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
38 simplll 815 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (SubMnd‘𝑊))
39 simpr 479 . . . . . . . . . . . 12 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
4039nnnn0d 11543 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41 simpllr 817 . . . . . . . . . . . 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 2842 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥𝐴)
44 eqid 2760 . . . . . . . . . . . 12 (.g‘(𝑊s 𝐴)) = (.g‘(𝑊s 𝐴))
454, 11, 44submmulg 17787 . . . . . . . . . . 11 ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0𝑥𝐴) → (𝑛(.g𝑊)𝑥) = (𝑛(.g‘(𝑊s 𝐴))𝑥))
4638, 40, 43, 45syl3anc 1477 . . . . . . . . . 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 4818 . . . . . . . . 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 3187 . . . . . . . 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 3123 . . . . . 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 3123 . . . . 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 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 29969 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Toset)
5627adantl 473 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Mnd)
57 eqid 2760 . . . . 5 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
58 eqid 2760 . . . . 5 (0g‘(𝑊s 𝐴)) = (0g‘(𝑊s 𝐴))
5957, 58, 44, 28, 29isarchi2 30048 . . . 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 696 . . 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 753 . 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 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ∖ cdif 3712   ⊆ wss 3715   class class class wbr 4804   I cid 5173  ‘cfv 6049  (class class class)co 6813  ℕcn 11212  ℕ0cn0 11484  Basecbs 16059   ↾s cress 16060  lecple 16150  0gc0g 16302  ltcplt 17142  Tosetctos 17234  Mndcmnd 17495  SubMndcsubmnd 17535  .gcmg 17741  Archicarchi 30040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-seq 12996  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-ple 16163  df-0g 16304  df-preset 17129  df-poset 17147  df-plt 17159  df-toset 17235  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-inftm 30041  df-archi 30042 This theorem is referenced by:  nn0archi  30152
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