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Theorem torsubg 19632
Description: The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.)
Hypothesis
Ref Expression
torsubg.1 𝑂 = (od‘𝐺)
Assertion
Ref Expression
torsubg (𝐺 ∈ Abel → (𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Proof of Theorem torsubg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 6033 . . . 4 (𝑂 “ ℕ) ⊆ dom 𝑂
2 eqid 2736 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 torsubg.1 . . . . . 6 𝑂 = (od‘𝐺)
42, 3odf 19319 . . . . 5 𝑂:(Base‘𝐺)⟶ℕ0
54fdmi 6680 . . . 4 dom 𝑂 = (Base‘𝐺)
61, 5sseqtri 3980 . . 3 (𝑂 “ ℕ) ⊆ (Base‘𝐺)
76a1i 11 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ⊆ (Base‘𝐺))
8 ablgrp 19567 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9 eqid 2736 . . . . . 6 (0g𝐺) = (0g𝐺)
102, 9grpidcl 18778 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
118, 10syl 17 . . . 4 (𝐺 ∈ Abel → (0g𝐺) ∈ (Base‘𝐺))
123, 9od1 19341 . . . . . 6 (𝐺 ∈ Grp → (𝑂‘(0g𝐺)) = 1)
138, 12syl 17 . . . . 5 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) = 1)
14 1nn 12164 . . . . 5 1 ∈ ℕ
1513, 14eqeltrdi 2846 . . . 4 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) ∈ ℕ)
16 ffn 6668 . . . . . 6 (𝑂:(Base‘𝐺)⟶ℕ0𝑂 Fn (Base‘𝐺))
174, 16ax-mp 5 . . . . 5 𝑂 Fn (Base‘𝐺)
18 elpreima 7008 . . . . 5 (𝑂 Fn (Base‘𝐺) → ((0g𝐺) ∈ (𝑂 “ ℕ) ↔ ((0g𝐺) ∈ (Base‘𝐺) ∧ (𝑂‘(0g𝐺)) ∈ ℕ)))
1917, 18ax-mp 5 . . . 4 ((0g𝐺) ∈ (𝑂 “ ℕ) ↔ ((0g𝐺) ∈ (Base‘𝐺) ∧ (𝑂‘(0g𝐺)) ∈ ℕ))
2011, 15, 19sylanbrc 583 . . 3 (𝐺 ∈ Abel → (0g𝐺) ∈ (𝑂 “ ℕ))
2120ne0d 4295 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ≠ ∅)
228ad2antrr 724 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Grp)
236sseli 3940 . . . . . . . 8 (𝑥 ∈ (𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
2423ad2antlr 725 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
256sseli 3940 . . . . . . . 8 (𝑦 ∈ (𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
2625adantl 482 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27 eqid 2736 . . . . . . . 8 (+g𝐺) = (+g𝐺)
282, 27grpcl 18756 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
2922, 24, 26, 28syl3anc 1371 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
30 0nnn 12189 . . . . . . . . 9 ¬ 0 ∈ ℕ
312, 3odcl 19318 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (Base‘𝐺) → (𝑂𝑥) ∈ ℕ0)
3224, 31syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ0)
3332nn0zd 12525 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℤ)
342, 3odcl 19318 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (Base‘𝐺) → (𝑂𝑦) ∈ ℕ0)
3526, 34syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ0)
3635nn0zd 12525 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℤ)
3733, 36gcdcld 16388 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℕ0)
3837nn0cnd 12475 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℂ)
3938mul02d 11353 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) = 0)
4039breq1d 5115 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ 0 ∥ ((𝑂𝑥) · (𝑂𝑦))))
4133, 36zmulcld 12613 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℤ)
42 0dvds 16159 . . . . . . . . . . . 12 (((𝑂𝑥) · (𝑂𝑦)) ∈ ℤ → (0 ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
4341, 42syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (0 ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
4440, 43bitrd 278 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
45 elpreima 7008 . . . . . . . . . . . . . . 15 (𝑂 Fn (Base‘𝐺) → (𝑥 ∈ (𝑂 “ ℕ) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝑂𝑥) ∈ ℕ)))
4617, 45ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑂 “ ℕ) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝑂𝑥) ∈ ℕ))
4746simprbi 497 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑂 “ ℕ) → (𝑂𝑥) ∈ ℕ)
4847ad2antlr 725 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
49 elpreima 7008 . . . . . . . . . . . . . . 15 (𝑂 Fn (Base‘𝐺) → (𝑦 ∈ (𝑂 “ ℕ) ↔ (𝑦 ∈ (Base‘𝐺) ∧ (𝑂𝑦) ∈ ℕ)))
5017, 49ax-mp 5 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑂 “ ℕ) ↔ (𝑦 ∈ (Base‘𝐺) ∧ (𝑂𝑦) ∈ ℕ))
5150simprbi 497 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑂 “ ℕ) → (𝑂𝑦) ∈ ℕ)
5251adantl 482 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ)
5348, 52nnmulcld 12206 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ)
54 eleq1 2825 . . . . . . . . . . 11 (((𝑂𝑥) · (𝑂𝑦)) = 0 → (((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
5553, 54syl5ibcom 244 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (((𝑂𝑥) · (𝑂𝑦)) = 0 → 0 ∈ ℕ))
5644, 55sylbid 239 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) → 0 ∈ ℕ))
5730, 56mtoi 198 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
58 simpll 765 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Abel)
593, 2, 27odadd1 19626 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
6058, 24, 26, 59syl3anc 1371 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
61 oveq1 7364 . . . . . . . . . 10 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) = (0 · ((𝑂𝑥) gcd (𝑂𝑦))))
6261breq1d 5115 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6360, 62syl5ibcom 244 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6457, 63mtod 197 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0)
652, 3odcl 19318 . . . . . . . . . 10 ((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0)
6629, 65syl 17 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0)
67 elnn0 12415 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0 ↔ ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
6866, 67sylib 217 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
6968ord 862 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (¬ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ → (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
7064, 69mt3d 148 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ)
71 elpreima 7008 . . . . . . 7 (𝑂 Fn (Base‘𝐺) → ((𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ↔ ((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ)))
7217, 71ax-mp 5 . . . . . 6 ((𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ↔ ((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ))
7329, 70, 72sylanbrc 583 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
7473ralrimiva 3143 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
75 eqid 2736 . . . . . . 7 (invg𝐺) = (invg𝐺)
762, 75grpinvcl 18798 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
778, 23, 76syl2an 596 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
783, 75, 2odinv 19343 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
798, 23, 78syl2an 596 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
8047adantl 482 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
8179, 80eqeltrd 2838 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ)
82 elpreima 7008 . . . . . 6 (𝑂 Fn (Base‘𝐺) → (((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ) ↔ (((invg𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ)))
8317, 82ax-mp 5 . . . . 5 (((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ) ↔ (((invg𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ))
8477, 81, 83sylanbrc 583 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ))
8574, 84jca 512 . . 3 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
8685ralrimiva 3143 . 2 (𝐺 ∈ Abel → ∀𝑥 ∈ (𝑂 “ ℕ)(∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
872, 27, 75issubg2 18943 . . 3 (𝐺 ∈ Grp → ((𝑂 “ ℕ) ∈ (SubGrp‘𝐺) ↔ ((𝑂 “ ℕ) ⊆ (Base‘𝐺) ∧ (𝑂 “ ℕ) ≠ ∅ ∧ ∀𝑥 ∈ (𝑂 “ ℕ)(∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))))
888, 87syl 17 . 2 (𝐺 ∈ Abel → ((𝑂 “ ℕ) ∈ (SubGrp‘𝐺) ↔ ((𝑂 “ ℕ) ⊆ (Base‘𝐺) ∧ (𝑂 “ ℕ) ≠ ∅ ∧ ∀𝑥 ∈ (𝑂 “ ℕ)(∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))))
897, 21, 86, 88mpbir3and 1342 1 (𝐺 ∈ Abel → (𝑂 “ ℕ) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wss 3910  c0 4282   class class class wbr 5105  ccnv 5632  dom cdm 5633  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   · cmul 11056  cn 12153  0cn0 12413  cz 12499  cdvds 16136   gcd cgcd 16374  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749  SubGrpcsubg 18922  odcod 19306  Abelcabl 19563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-gcd 16375  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-od 19310  df-cmn 19564  df-abl 19565
This theorem is referenced by: (None)
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