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Theorem torsubg 19766
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 6071 . . . 4 (𝑂 “ ℕ) ⊆ dom 𝑂
2 eqid 2724 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 torsubg.1 . . . . . 6 𝑂 = (od‘𝐺)
42, 3odf 19449 . . . . 5 𝑂:(Base‘𝐺)⟶ℕ0
54fdmi 6720 . . . 4 dom 𝑂 = (Base‘𝐺)
61, 5sseqtri 4011 . . 3 (𝑂 “ ℕ) ⊆ (Base‘𝐺)
76a1i 11 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ⊆ (Base‘𝐺))
8 ablgrp 19697 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9 eqid 2724 . . . . . 6 (0g𝐺) = (0g𝐺)
102, 9grpidcl 18887 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
118, 10syl 17 . . . 4 (𝐺 ∈ Abel → (0g𝐺) ∈ (Base‘𝐺))
123, 9od1 19471 . . . . . 6 (𝐺 ∈ Grp → (𝑂‘(0g𝐺)) = 1)
138, 12syl 17 . . . . 5 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) = 1)
14 1nn 12221 . . . . 5 1 ∈ ℕ
1513, 14eqeltrdi 2833 . . . 4 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) ∈ ℕ)
16 ffn 6708 . . . . . 6 (𝑂:(Base‘𝐺)⟶ℕ0𝑂 Fn (Base‘𝐺))
174, 16ax-mp 5 . . . . 5 𝑂 Fn (Base‘𝐺)
18 elpreima 7050 . . . . 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 582 . . 3 (𝐺 ∈ Abel → (0g𝐺) ∈ (𝑂 “ ℕ))
2120ne0d 4328 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ≠ ∅)
228ad2antrr 723 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Grp)
236sseli 3971 . . . . . . . 8 (𝑥 ∈ (𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
2423ad2antlr 724 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
256sseli 3971 . . . . . . . 8 (𝑦 ∈ (𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
2625adantl 481 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27 eqid 2724 . . . . . . . 8 (+g𝐺) = (+g𝐺)
282, 27grpcl 18863 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
2922, 24, 26, 28syl3anc 1368 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
30 0nnn 12246 . . . . . . . . 9 ¬ 0 ∈ ℕ
312, 3odcl 19448 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (Base‘𝐺) → (𝑂𝑥) ∈ ℕ0)
3224, 31syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ0)
3332nn0zd 12582 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℤ)
342, 3odcl 19448 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (Base‘𝐺) → (𝑂𝑦) ∈ ℕ0)
3526, 34syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ0)
3635nn0zd 12582 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℤ)
3733, 36gcdcld 16448 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℕ0)
3837nn0cnd 12532 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℂ)
3938mul02d 11410 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) = 0)
4039breq1d 5149 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ 0 ∥ ((𝑂𝑥) · (𝑂𝑦))))
4133, 36zmulcld 12670 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℤ)
42 0dvds 16219 . . . . . . . . . . . 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 279 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
45 elpreima 7050 . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑂 “ ℕ) → (𝑂𝑥) ∈ ℕ)
4847ad2antlr 724 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
49 elpreima 7050 . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑂 “ ℕ) → (𝑂𝑦) ∈ ℕ)
5251adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ)
5348, 52nnmulcld 12263 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ)
54 eleq1 2813 . . . . . . . . . . 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 764 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Abel)
593, 2, 27odadd1 19760 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
6058, 24, 26, 59syl3anc 1368 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
61 oveq1 7409 . . . . . . . . . 10 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) = (0 · ((𝑂𝑥) gcd (𝑂𝑦))))
6261breq1d 5149 . . . . . . . . 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 19448 . . . . . . . . . 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 12472 . . . . . . . . 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 861 . . . . . . 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 7050 . . . . . . 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 582 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
7473ralrimiva 3138 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
75 eqid 2724 . . . . . . 7 (invg𝐺) = (invg𝐺)
762, 75grpinvcl 18909 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
778, 23, 76syl2an 595 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
783, 75, 2odinv 19473 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
798, 23, 78syl2an 595 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
8047adantl 481 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
8179, 80eqeltrd 2825 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ)
82 elpreima 7050 . . . . . 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 582 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ))
8574, 84jca 511 . . 3 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
8685ralrimiva 3138 . 2 (𝐺 ∈ Abel → ∀𝑥 ∈ (𝑂 “ ℕ)(∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
872, 27, 75issubg2 19060 . . 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 1339 1 (𝐺 ∈ Abel → (𝑂 “ ℕ) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wral 3053  wss 3941  c0 4315   class class class wbr 5139  ccnv 5666  dom cdm 5667  cima 5670   Fn wfn 6529  wf 6530  cfv 6534  (class class class)co 7402  0cc0 11107  1c1 11108   · cmul 11112  cn 12210  0cn0 12470  cz 12556  cdvds 16196   gcd cgcd 16434  Basecbs 17145  +gcplusg 17198  0gc0g 17386  Grpcgrp 18855  invgcminusg 18856  SubGrpcsubg 19039  odcod 19436  Abelcabl 19693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-inf 9435  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-div 11870  df-nn 12211  df-2 12273  df-3 12274  df-n0 12471  df-z 12557  df-uz 12821  df-rp 12973  df-fz 13483  df-fzo 13626  df-fl 13755  df-mod 13833  df-seq 13965  df-exp 14026  df-cj 15044  df-re 15045  df-im 15046  df-sqrt 15180  df-abs 15181  df-dvds 16197  df-gcd 16435  df-sets 17098  df-slot 17116  df-ndx 17128  df-base 17146  df-ress 17175  df-plusg 17211  df-0g 17388  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18988  df-subg 19042  df-od 19440  df-cmn 19694  df-abl 19695
This theorem is referenced by: (None)
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