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Theorem torsubg 18178
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 5444 . . . 4 (𝑂 “ ℕ) ⊆ dom 𝑂
2 eqid 2621 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 torsubg.1 . . . . . 6 𝑂 = (od‘𝐺)
42, 3odf 17877 . . . . 5 𝑂:(Base‘𝐺)⟶ℕ0
54fdmi 6009 . . . 4 dom 𝑂 = (Base‘𝐺)
61, 5sseqtri 3616 . . 3 (𝑂 “ ℕ) ⊆ (Base‘𝐺)
76a1i 11 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ⊆ (Base‘𝐺))
8 ablgrp 18119 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9 eqid 2621 . . . . . 6 (0g𝐺) = (0g𝐺)
102, 9grpidcl 17371 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
118, 10syl 17 . . . 4 (𝐺 ∈ Abel → (0g𝐺) ∈ (Base‘𝐺))
123, 9od1 17897 . . . . . 6 (𝐺 ∈ Grp → (𝑂‘(0g𝐺)) = 1)
138, 12syl 17 . . . . 5 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) = 1)
14 1nn 10975 . . . . 5 1 ∈ ℕ
1513, 14syl6eqel 2706 . . . 4 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) ∈ ℕ)
16 ffn 6002 . . . . . 6 (𝑂:(Base‘𝐺)⟶ℕ0𝑂 Fn (Base‘𝐺))
174, 16ax-mp 5 . . . . 5 𝑂 Fn (Base‘𝐺)
18 elpreima 6293 . . . . 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 697 . . 3 (𝐺 ∈ Abel → (0g𝐺) ∈ (𝑂 “ ℕ))
21 ne0i 3897 . . 3 ((0g𝐺) ∈ (𝑂 “ ℕ) → (𝑂 “ ℕ) ≠ ∅)
2220, 21syl 17 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ≠ ∅)
238ad2antrr 761 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Grp)
246sseli 3579 . . . . . . . 8 (𝑥 ∈ (𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
2524ad2antlr 762 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
266sseli 3579 . . . . . . . 8 (𝑦 ∈ (𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
2726adantl 482 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
28 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
292, 28grpcl 17351 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
3023, 25, 27, 29syl3anc 1323 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
31 0nnn 10996 . . . . . . . . 9 ¬ 0 ∈ ℕ
322, 3odcl 17876 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (Base‘𝐺) → (𝑂𝑥) ∈ ℕ0)
3325, 32syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ0)
3433nn0zd 11424 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℤ)
352, 3odcl 17876 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (Base‘𝐺) → (𝑂𝑦) ∈ ℕ0)
3627, 35syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ0)
3736nn0zd 11424 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℤ)
3834, 37gcdcld 15154 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℕ0)
3938nn0cnd 11297 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℂ)
4039mul02d 10178 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) = 0)
4140breq1d 4623 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ 0 ∥ ((𝑂𝑥) · (𝑂𝑦))))
4234, 37zmulcld 11432 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℤ)
43 0dvds 14926 . . . . . . . . . . . 12 (((𝑂𝑥) · (𝑂𝑦)) ∈ ℤ → (0 ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
4442, 43syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (0 ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
4541, 44bitrd 268 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
46 elpreima 6293 . . . . . . . . . . . . . . 15 (𝑂 Fn (Base‘𝐺) → (𝑥 ∈ (𝑂 “ ℕ) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝑂𝑥) ∈ ℕ)))
4717, 46ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑂 “ ℕ) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝑂𝑥) ∈ ℕ))
4847simprbi 480 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑂 “ ℕ) → (𝑂𝑥) ∈ ℕ)
4948ad2antlr 762 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
50 elpreima 6293 . . . . . . . . . . . . . . 15 (𝑂 Fn (Base‘𝐺) → (𝑦 ∈ (𝑂 “ ℕ) ↔ (𝑦 ∈ (Base‘𝐺) ∧ (𝑂𝑦) ∈ ℕ)))
5117, 50ax-mp 5 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑂 “ ℕ) ↔ (𝑦 ∈ (Base‘𝐺) ∧ (𝑂𝑦) ∈ ℕ))
5251simprbi 480 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑂 “ ℕ) → (𝑂𝑦) ∈ ℕ)
5352adantl 482 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ)
5449, 53nnmulcld 11012 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ)
55 eleq1 2686 . . . . . . . . . . 11 (((𝑂𝑥) · (𝑂𝑦)) = 0 → (((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
5654, 55syl5ibcom 235 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (((𝑂𝑥) · (𝑂𝑦)) = 0 → 0 ∈ ℕ))
5745, 56sylbid 230 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) → 0 ∈ ℕ))
5831, 57mtoi 190 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
59 simpll 789 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Abel)
603, 2, 28odadd1 18172 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
6159, 25, 27, 60syl3anc 1323 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
62 oveq1 6611 . . . . . . . . . 10 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) = (0 · ((𝑂𝑥) gcd (𝑂𝑦))))
6362breq1d 4623 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6461, 63syl5ibcom 235 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6558, 64mtod 189 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0)
662, 3odcl 17876 . . . . . . . . . 10 ((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0)
6730, 66syl 17 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0)
68 elnn0 11238 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0 ↔ ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
6967, 68sylib 208 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
7069ord 392 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (¬ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ → (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
7165, 70mt3d 140 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ)
72 elpreima 6293 . . . . . . 7 (𝑂 Fn (Base‘𝐺) → ((𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ↔ ((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ)))
7317, 72ax-mp 5 . . . . . 6 ((𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ↔ ((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ))
7430, 71, 73sylanbrc 697 . . . . 5 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
7574ralrimiva 2960 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
76 eqid 2621 . . . . . . 7 (invg𝐺) = (invg𝐺)
772, 76grpinvcl 17388 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
788, 24, 77syl2an 494 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (Base‘𝐺))
793, 76, 2odinv 17899 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
808, 24, 79syl2an 494 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) = (𝑂𝑥))
8148adantl 482 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
8280, 81eqeltrd 2698 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ)
83 elpreima 6293 . . . . . 6 (𝑂 Fn (Base‘𝐺) → (((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ) ↔ (((invg𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ)))
8417, 83ax-mp 5 . . . . 5 (((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ) ↔ (((invg𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ))
8578, 82, 84sylanbrc 697 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ))
8675, 85jca 554 . . 3 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
8786ralrimiva 2960 . 2 (𝐺 ∈ Abel → ∀𝑥 ∈ (𝑂 “ ℕ)(∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
882, 28, 76issubg2 17530 . . 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)))))
898, 88syl 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)))))
907, 22, 87, 89mpbir3and 1243 1 (𝐺 ∈ Abel → (𝑂 “ ℕ) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wss 3555  c0 3891   class class class wbr 4613  ccnv 5073  dom cdm 5074  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  0cc0 9880  1c1 9881   · cmul 9885  cn 10964  0cn0 11236  cz 11321  cdvds 14907   gcd cgcd 15140  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343  invgcminusg 17344  SubGrpcsubg 17509  odcod 17865  Abelcabl 18115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-dvds 14908  df-gcd 15141  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-od 17869  df-cmn 18116  df-abl 18117
This theorem is referenced by: (None)
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