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Theorem torsubg 18537
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 5669 . . . 4 (𝑂 “ ℕ) ⊆ dom 𝑂
2 eqid 2765 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 torsubg.1 . . . . . 6 𝑂 = (od‘𝐺)
42, 3odf 18234 . . . . 5 𝑂:(Base‘𝐺)⟶ℕ0
54fdmi 6235 . . . 4 dom 𝑂 = (Base‘𝐺)
61, 5sseqtri 3799 . . 3 (𝑂 “ ℕ) ⊆ (Base‘𝐺)
76a1i 11 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ⊆ (Base‘𝐺))
8 ablgrp 18478 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9 eqid 2765 . . . . . 6 (0g𝐺) = (0g𝐺)
102, 9grpidcl 17731 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
118, 10syl 17 . . . 4 (𝐺 ∈ Abel → (0g𝐺) ∈ (Base‘𝐺))
123, 9od1 18254 . . . . . 6 (𝐺 ∈ Grp → (𝑂‘(0g𝐺)) = 1)
138, 12syl 17 . . . . 5 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) = 1)
14 1nn 11291 . . . . 5 1 ∈ ℕ
1513, 14syl6eqel 2852 . . . 4 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) ∈ ℕ)
16 ffn 6225 . . . . . 6 (𝑂:(Base‘𝐺)⟶ℕ0𝑂 Fn (Base‘𝐺))
174, 16ax-mp 5 . . . . 5 𝑂 Fn (Base‘𝐺)
18 elpreima 6531 . . . . 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 578 . . 3 (𝐺 ∈ Abel → (0g𝐺) ∈ (𝑂 “ ℕ))
2120ne0d 4088 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ≠ ∅)
228ad2antrr 717 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Grp)
236sseli 3759 . . . . . . . 8 (𝑥 ∈ (𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
2423ad2antlr 718 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
256sseli 3759 . . . . . . . 8 (𝑦 ∈ (𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
2625adantl 473 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27 eqid 2765 . . . . . . . 8 (+g𝐺) = (+g𝐺)
282, 27grpcl 17711 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
2922, 24, 26, 28syl3anc 1490 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
30 0nnn 11313 . . . . . . . . 9 ¬ 0 ∈ ℕ
312, 3odcl 18233 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (Base‘𝐺) → (𝑂𝑥) ∈ ℕ0)
3224, 31syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ0)
3332nn0zd 11732 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℤ)
342, 3odcl 18233 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (Base‘𝐺) → (𝑂𝑦) ∈ ℕ0)
3526, 34syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ0)
3635nn0zd 11732 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℤ)
3733, 36gcdcld 15525 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℕ0)
3837nn0cnd 11604 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℂ)
3938mul02d 10492 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) = 0)
4039breq1d 4821 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ 0 ∥ ((𝑂𝑥) · (𝑂𝑦))))
4133, 36zmulcld 11740 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℤ)
42 0dvds 15301 . . . . . . . . . . . 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 270 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
45 elpreima 6531 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑂 “ ℕ) → (𝑂𝑥) ∈ ℕ)
4847ad2antlr 718 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
49 elpreima 6531 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑂 “ ℕ) → (𝑂𝑦) ∈ ℕ)
5251adantl 473 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ)
5348, 52nnmulcld 11329 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ)
54 eleq1 2832 . . . . . . . . . . 11 (((𝑂𝑥) · (𝑂𝑦)) = 0 → (((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
5553, 54syl5ibcom 236 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (((𝑂𝑥) · (𝑂𝑦)) = 0 → 0 ∈ ℕ))
5644, 55sylbid 231 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) → 0 ∈ ℕ))
5730, 56mtoi 190 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
58 simpll 783 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Abel)
593, 2, 27odadd1 18531 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
6058, 24, 26, 59syl3anc 1490 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
61 oveq1 6853 . . . . . . . . . 10 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) = (0 · ((𝑂𝑥) gcd (𝑂𝑦))))
6261breq1d 4821 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6360, 62syl5ibcom 236 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6457, 63mtod 189 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0)
652, 3odcl 18233 . . . . . . . . . 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 11544 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0 ↔ ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
6866, 67sylib 209 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
6968ord 890 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (¬ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ → (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
7064, 69mt3d 142 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ)
71 elpreima 6531 . . . . . . 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 578 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
7473ralrimiva 3113 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
75 eqid 2765 . . . . . . 7 (invg𝐺) = (invg𝐺)
762, 75grpinvcl 17748 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
778, 23, 76syl2an 589 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
783, 75, 2odinv 18256 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
798, 23, 78syl2an 589 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
8047adantl 473 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
8179, 80eqeltrd 2844 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ)
82 elpreima 6531 . . . . . 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 578 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ))
8574, 84jca 507 . . 3 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
8685ralrimiva 3113 . 2 (𝐺 ∈ Abel → ∀𝑥 ∈ (𝑂 “ ℕ)(∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
872, 27, 75issubg2 17887 . . 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 1442 1 (𝐺 ∈ Abel → (𝑂 “ ℕ) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wss 3734  c0 4081   class class class wbr 4811  ccnv 5278  dom cdm 5279  cima 5282   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6846  0cc0 10193  1c1 10194   · cmul 10198  cn 11278  0cn0 11542  cz 11628  cdvds 15279   gcd cgcd 15511  Basecbs 16144  +gcplusg 16228  0gc0g 16380  Grpcgrp 17703  invgcminusg 17704  SubGrpcsubg 17866  odcod 18222  Abelcabl 18474
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 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-sup 8559  df-inf 8560  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-z 11629  df-uz 11892  df-rp 12034  df-fz 12539  df-fzo 12679  df-fl 12806  df-mod 12882  df-seq 13014  df-exp 13073  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-dvds 15280  df-gcd 15512  df-ndx 16147  df-slot 16148  df-base 16150  df-sets 16151  df-ress 16152  df-plusg 16241  df-0g 16382  df-mgm 17522  df-sgrp 17564  df-mnd 17575  df-grp 17706  df-minusg 17707  df-sbg 17708  df-mulg 17822  df-subg 17869  df-od 18226  df-cmn 18475  df-abl 18476
This theorem is referenced by: (None)
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