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Theorem torsubg 18894
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 5947 . . . 4 (𝑂 “ ℕ) ⊆ dom 𝑂
2 eqid 2826 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 torsubg.1 . . . . . 6 𝑂 = (od‘𝐺)
42, 3odf 18585 . . . . 5 𝑂:(Base‘𝐺)⟶ℕ0
54fdmi 6521 . . . 4 dom 𝑂 = (Base‘𝐺)
61, 5sseqtri 4007 . . 3 (𝑂 “ ℕ) ⊆ (Base‘𝐺)
76a1i 11 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ⊆ (Base‘𝐺))
8 ablgrp 18831 . . . . 5 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9 eqid 2826 . . . . . 6 (0g𝐺) = (0g𝐺)
102, 9grpidcl 18061 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
118, 10syl 17 . . . 4 (𝐺 ∈ Abel → (0g𝐺) ∈ (Base‘𝐺))
123, 9od1 18606 . . . . . 6 (𝐺 ∈ Grp → (𝑂‘(0g𝐺)) = 1)
138, 12syl 17 . . . . 5 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) = 1)
14 1nn 11638 . . . . 5 1 ∈ ℕ
1513, 14syl6eqel 2926 . . . 4 (𝐺 ∈ Abel → (𝑂‘(0g𝐺)) ∈ ℕ)
16 ffn 6511 . . . . . 6 (𝑂:(Base‘𝐺)⟶ℕ0𝑂 Fn (Base‘𝐺))
174, 16ax-mp 5 . . . . 5 𝑂 Fn (Base‘𝐺)
18 elpreima 6824 . . . . 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 4305 . 2 (𝐺 ∈ Abel → (𝑂 “ ℕ) ≠ ∅)
228ad2antrr 722 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Grp)
236sseli 3967 . . . . . . . 8 (𝑥 ∈ (𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
2423ad2antlr 723 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
256sseli 3967 . . . . . . . 8 (𝑦 ∈ (𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
2625adantl 482 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27 eqid 2826 . . . . . . . 8 (+g𝐺) = (+g𝐺)
282, 27grpcl 18041 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
2922, 24, 26, 28syl3anc 1365 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
30 0nnn 11662 . . . . . . . . 9 ¬ 0 ∈ ℕ
312, 3odcl 18584 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (Base‘𝐺) → (𝑂𝑥) ∈ ℕ0)
3224, 31syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ0)
3332nn0zd 12074 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℤ)
342, 3odcl 18584 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (Base‘𝐺) → (𝑂𝑦) ∈ ℕ0)
3526, 34syl 17 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℕ0)
3635nn0zd 12074 . . . . . . . . . . . . . . 15 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑦) ∈ ℤ)
3733, 36gcdcld 15847 . . . . . . . . . . . . . 14 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℕ0)
3837nn0cnd 11946 . . . . . . . . . . . . 13 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) gcd (𝑂𝑦)) ∈ ℂ)
3938mul02d 10827 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) = 0)
4039breq1d 5073 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ 0 ∥ ((𝑂𝑥) · (𝑂𝑦))))
4133, 36zmulcld 12082 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℤ)
42 0dvds 15620 . . . . . . . . . . . 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 280 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ ((𝑂𝑥) · (𝑂𝑦)) = 0))
45 elpreima 6824 . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . 12 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
49 elpreima 6824 . . . . . . . . . . . . . . 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 11679 . . . . . . . . . . 11 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ)
54 eleq1 2905 . . . . . . . . . . 11 (((𝑂𝑥) · (𝑂𝑦)) = 0 → (((𝑂𝑥) · (𝑂𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
5553, 54syl5ibcom 246 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (((𝑂𝑥) · (𝑂𝑦)) = 0 → 0 ∈ ℕ))
5644, 55sylbid 241 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) → 0 ∈ ℕ))
5730, 56mtoi 200 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
58 simpll 763 . . . . . . . . . 10 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → 𝐺 ∈ Abel)
593, 2, 27odadd1 18888 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
6058, 24, 26, 59syl3anc 1365 . . . . . . . . 9 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)))
61 oveq1 7155 . . . . . . . . . 10 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → ((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) = (0 · ((𝑂𝑥) gcd (𝑂𝑦))))
6261breq1d 5073 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (((𝑂‘(𝑥(+g𝐺)𝑦)) · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦)) ↔ (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6360, 62syl5ibcom 246 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) = 0 → (0 · ((𝑂𝑥) gcd (𝑂𝑦))) ∥ ((𝑂𝑥) · (𝑂𝑦))))
6457, 63mtod 199 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0)
652, 3odcl 18584 . . . . . . . . . 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 11888 . . . . . . . . 9 ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ0 ↔ ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
6866, 67sylib 219 . . . . . . . 8 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
6968ord 860 . . . . . . 7 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (¬ (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ → (𝑂‘(𝑥(+g𝐺)𝑦)) = 0))
7064, 69mt3d 150 . . . . . 6 (((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) ∧ 𝑦 ∈ (𝑂 “ ℕ)) → (𝑂‘(𝑥(+g𝐺)𝑦)) ∈ ℕ)
71 elpreima 6824 . . . . . . 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 3187 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → ∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ))
75 eqid 2826 . . . . . . 7 (invg𝐺) = (invg𝐺)
762, 75grpinvcl 18081 . . . . . 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 18608 . . . . . . 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 482 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂𝑥) ∈ ℕ)
8179, 80eqeltrd 2918 . . . . 5 ((𝐺 ∈ Abel ∧ 𝑥 ∈ (𝑂 “ ℕ)) → (𝑂‘((invg𝐺)‘𝑥)) ∈ ℕ)
82 elpreima 6824 . . . . . 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 3187 . 2 (𝐺 ∈ Abel → ∀𝑥 ∈ (𝑂 “ ℕ)(∀𝑦 ∈ (𝑂 “ ℕ)(𝑥(+g𝐺)𝑦) ∈ (𝑂 “ ℕ) ∧ ((invg𝐺)‘𝑥) ∈ (𝑂 “ ℕ)))
872, 27, 75issubg2 18224 . . 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 1336 1 (𝐺 ∈ Abel → (𝑂 “ ℕ) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wss 3940  c0 4295   class class class wbr 5063  ccnv 5553  dom cdm 5554  cima 5557   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  0cc0 10526  1c1 10527   · cmul 10531  cn 11627  0cn0 11886  cz 11970  cdvds 15597   gcd cgcd 15833  Basecbs 16473  +gcplusg 16555  0gc0g 16703  Grpcgrp 18033  invgcminusg 18034  SubGrpcsubg 18203  odcod 18572  Abelcabl 18827
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-fl 13152  df-mod 13228  df-seq 13360  df-exp 13420  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-dvds 15598  df-gcd 15834  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-0g 16705  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-grp 18036  df-minusg 18037  df-sbg 18038  df-mulg 18155  df-subg 18206  df-od 18576  df-cmn 18828  df-abl 18829
This theorem is referenced by: (None)
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