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Theorem submarchi 33193
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 18815 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → 𝑊 ∈ Mnd)
2 eqid 2737 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
3 eqid 2737 . . . . . . 7 (0g𝑊) = (0g𝑊)
4 eqid 2737 . . . . . . 7 (.g𝑊) = (.g𝑊)
5 eqid 2737 . . . . . . 7 (le‘𝑊) = (le‘𝑊)
6 eqid 2737 . . . . . . 7 (lt‘𝑊) = (lt‘𝑊)
72, 3, 4, 5, 6isarchi2 33192 . . . . . 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 593 . . . . 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 476 . . . 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 652 . . 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 2737 . . . . . . . 8 (𝑊s 𝐴) = (𝑊s 𝐴)
1211submbas 18827 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 = (Base‘(𝑊s 𝐴)))
132submss 18822 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
1412, 13eqsstrrd 4019 . . . . . 6 (𝐴 ∈ (SubMnd‘𝑊) → (Base‘(𝑊s 𝐴)) ⊆ (Base‘𝑊))
15 ssralv 4052 . . . . . . . 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 3169 . . . . . . 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 4052 . . . . . . 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 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 18828 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2221ad2antrr 726 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (0g𝑊) = (0g‘(𝑊s 𝐴)))
2311, 5ressle 17424 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
2423difeq1d 4125 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → ((le‘𝑊) ∖ I ) = ((le‘(𝑊s 𝐴)) ∖ I ))
255, 6pltfval 18376 . . . . . . . . . . . 12 (𝑊 ∈ Mnd → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
261, 25syl 17 . . . . . . . . . . 11 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = ((le‘𝑊) ∖ I ))
2711submmnd 18826 . . . . . . . . . . . 12 (𝐴 ∈ (SubMnd‘𝑊) → (𝑊s 𝐴) ∈ Mnd)
28 eqid 2737 . . . . . . . . . . . . 13 (le‘(𝑊s 𝐴)) = (le‘(𝑊s 𝐴))
29 eqid 2737 . . . . . . . . . . . . 13 (lt‘(𝑊s 𝐴)) = (lt‘(𝑊s 𝐴))
3028, 29pltfval 18376 . . . . . . . . . . . 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 2787 . . . . . . . . . 10 (𝐴 ∈ (SubMnd‘𝑊) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
3332ad2antrr 726 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → (lt‘𝑊) = (lt‘(𝑊s 𝐴)))
34 eqidd 2738 . . . . . . . . 9 (((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) → 𝑥 = 𝑥)
3522, 33, 34breq123d 5157 . . . . . . . 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 2738 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑦)
3723ad3antrrr 730 . . . . . . . . . 10 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → (le‘𝑊) = (le‘(𝑊s 𝐴)))
38 simplll 775 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (SubMnd‘𝑊))
39 simpr 484 . . . . . . . . . . . 12 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
4039nnnn0d 12587 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
41 simpllr 776 . . . . . . . . . . . 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 2844 . . . . . . . . . . 11 ((((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑥 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑦 ∈ (Base‘(𝑊s 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝑥𝐴)
44 eqid 2737 . . . . . . . . . . . 12 (.g‘(𝑊s 𝐴)) = (.g‘(𝑊s 𝐴))
454, 11, 44submmulg 19136 . . . . . . . . . . 11 ((𝐴 ∈ (SubMnd‘𝑊) ∧ 𝑛 ∈ ℕ0𝑥𝐴) → (𝑛(.g𝑊)𝑥) = (𝑛(.g‘(𝑊s 𝐴))𝑥))
4638, 40, 43, 45syl3anc 1373 . . . . . . . . . 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 5157 . . . . . . . . 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 3177 . . . . . . . 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 344 . . . . . . 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 3176 . . . . . 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 3176 . . . . 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 239 . . 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 32957 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Toset)
5627adantl 481 . . . 4 ((𝑊 ∈ Toset ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Mnd)
57 eqid 2737 . . . . 5 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
58 eqid 2737 . . . . 5 (0g‘(𝑊s 𝐴)) = (0g‘(𝑊s 𝐴))
5957, 58, 44, 28, 29isarchi2 33192 . . . 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 584 . . 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 715 . 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 257 1 (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cdif 3948  wss 3951   class class class wbr 5143   I cid 5577  cfv 6561  (class class class)co 7431  cn 12266  0cn0 12526  Basecbs 17247  s cress 17274  lecple 17304  0gc0g 17484  ltcplt 18354  Tosetctos 18461  Mndcmnd 18747  SubMndcsubmnd 18795  .gcmg 19085  Archicarchi 33184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-seq 14043  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-ple 17317  df-0g 17486  df-proset 18340  df-poset 18359  df-plt 18375  df-toset 18462  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-inftm 33185  df-archi 33186
This theorem is referenced by:  nn0archi  33375
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