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Theorem cycsubgcl 17841
Description: The set of integer powers of an element 𝐴 of a group forms a subgroup containing 𝐴, called the cyclic group generated by the element 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
cycsubg.x 𝑋 = (Base‘𝐺)
cycsubg.t · = (.g𝐺)
cycsubg.f 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
Assertion
Ref Expression
cycsubgcl ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥, ·   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem cycsubgcl
Dummy variables 𝑚 𝑛 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cycsubg.x . . . . . . . 8 𝑋 = (Base‘𝐺)
2 cycsubg.t . . . . . . . 8 · = (.g𝐺)
31, 2mulgcl 17780 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
433expa 1112 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
54an32s 881 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
6 cycsubg.f . . . . 5 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
75, 6fmptd 6549 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:ℤ⟶𝑋)
8 frn 6214 . . . 4 (𝐹:ℤ⟶𝑋 → ran 𝐹𝑋)
97, 8syl 17 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹𝑋)
10 1z 11619 . . . . . . 7 1 ∈ ℤ
11 oveq1 6821 . . . . . . . 8 (𝑥 = 1 → (𝑥 · 𝐴) = (1 · 𝐴))
12 ovex 6842 . . . . . . . 8 (1 · 𝐴) ∈ V
1311, 6, 12fvmpt 6445 . . . . . . 7 (1 ∈ ℤ → (𝐹‘1) = (1 · 𝐴))
1410, 13ax-mp 5 . . . . . 6 (𝐹‘1) = (1 · 𝐴)
151, 2mulg1 17769 . . . . . . 7 (𝐴𝑋 → (1 · 𝐴) = 𝐴)
1615adantl 473 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (1 · 𝐴) = 𝐴)
1714, 16syl5eq 2806 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) = 𝐴)
18 ffn 6206 . . . . . . 7 (𝐹:ℤ⟶𝑋𝐹 Fn ℤ)
197, 18syl 17 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹 Fn ℤ)
20 fnfvelrn 6520 . . . . . 6 ((𝐹 Fn ℤ ∧ 1 ∈ ℤ) → (𝐹‘1) ∈ ran 𝐹)
2119, 10, 20sylancl 697 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) ∈ ran 𝐹)
2217, 21eqeltrrd 2840 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐴 ∈ ran 𝐹)
23 ne0i 4064 . . . 4 (𝐴 ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
2422, 23syl 17 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ≠ ∅)
25 df-3an 1074 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋))
26 eqid 2760 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
271, 2, 26mulgdir 17794 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
2825, 27sylan2br 494 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
2928anass1rs 884 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
30 zaddcl 11629 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 + 𝑛) ∈ ℤ)
3130adantl 473 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) ∈ ℤ)
32 oveq1 6821 . . . . . . . . . . . . 13 (𝑥 = (𝑚 + 𝑛) → (𝑥 · 𝐴) = ((𝑚 + 𝑛) · 𝐴))
33 ovex 6842 . . . . . . . . . . . . 13 ((𝑚 + 𝑛) · 𝐴) ∈ V
3432, 6, 33fvmpt 6445 . . . . . . . . . . . 12 ((𝑚 + 𝑛) ∈ ℤ → (𝐹‘(𝑚 + 𝑛)) = ((𝑚 + 𝑛) · 𝐴))
3531, 34syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝑚 + 𝑛) · 𝐴))
36 oveq1 6821 . . . . . . . . . . . . . 14 (𝑥 = 𝑚 → (𝑥 · 𝐴) = (𝑚 · 𝐴))
37 ovex 6842 . . . . . . . . . . . . . 14 (𝑚 · 𝐴) ∈ V
3836, 6, 37fvmpt 6445 . . . . . . . . . . . . 13 (𝑚 ∈ ℤ → (𝐹𝑚) = (𝑚 · 𝐴))
3938ad2antrl 766 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑚) = (𝑚 · 𝐴))
40 oveq1 6821 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (𝑥 · 𝐴) = (𝑛 · 𝐴))
41 ovex 6842 . . . . . . . . . . . . . 14 (𝑛 · 𝐴) ∈ V
4240, 6, 41fvmpt 6445 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (𝐹𝑛) = (𝑛 · 𝐴))
4342ad2antll 767 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑛) = (𝑛 · 𝐴))
4439, 43oveq12d 6832 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
4529, 35, 443eqtr4d 2804 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
46 fnfvelrn 6520 . . . . . . . . . . 11 ((𝐹 Fn ℤ ∧ (𝑚 + 𝑛) ∈ ℤ) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
4719, 30, 46syl2an 495 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
4845, 47eqeltrrd 2840 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
4948anassrs 683 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) ∧ 𝑛 ∈ ℤ) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
5049ralrimiva 3104 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
51 oveq2 6822 . . . . . . . . . . 11 (𝑣 = (𝐹𝑛) → ((𝐹𝑚)(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
5251eleq1d 2824 . . . . . . . . . 10 (𝑣 = (𝐹𝑛) → (((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
5352ralrn 6526 . . . . . . . . 9 (𝐹 Fn ℤ → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
5419, 53syl 17 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
5554adantr 472 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
5650, 55mpbird 247 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹)
57 eqid 2760 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
581, 2, 57mulgneg 17781 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
59583expa 1112 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ) ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
6059an32s 881 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
61 znegcl 11624 . . . . . . . . . 10 (𝑚 ∈ ℤ → -𝑚 ∈ ℤ)
6261adantl 473 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → -𝑚 ∈ ℤ)
63 oveq1 6821 . . . . . . . . . 10 (𝑥 = -𝑚 → (𝑥 · 𝐴) = (-𝑚 · 𝐴))
64 ovex 6842 . . . . . . . . . 10 (-𝑚 · 𝐴) ∈ V
6563, 6, 64fvmpt 6445 . . . . . . . . 9 (-𝑚 ∈ ℤ → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
6662, 65syl 17 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
6738adantl 473 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹𝑚) = (𝑚 · 𝐴))
6867fveq2d 6357 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) = ((invg𝐺)‘(𝑚 · 𝐴)))
6960, 66, 683eqtr4d 2804 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = ((invg𝐺)‘(𝐹𝑚)))
70 fnfvelrn 6520 . . . . . . . 8 ((𝐹 Fn ℤ ∧ -𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
7119, 61, 70syl2an 495 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
7269, 71eqeltrrd 2840 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹)
7356, 72jca 555 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
7473ralrimiva 3104 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
75 oveq1 6821 . . . . . . . . 9 (𝑢 = (𝐹𝑚) → (𝑢(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)𝑣))
7675eleq1d 2824 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
7776ralbidv 3124 . . . . . . 7 (𝑢 = (𝐹𝑚) → (∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
78 fveq2 6353 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((invg𝐺)‘𝑢) = ((invg𝐺)‘(𝐹𝑚)))
7978eleq1d 2824 . . . . . . 7 (𝑢 = (𝐹𝑚) → (((invg𝐺)‘𝑢) ∈ ran 𝐹 ↔ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
8077, 79anbi12d 749 . . . . . 6 (𝑢 = (𝐹𝑚) → ((∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹) ↔ (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹)))
8180ralrn 6526 . . . . 5 (𝐹 Fn ℤ → (∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹) ↔ ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹)))
8219, 81syl 17 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹) ↔ ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹)))
8374, 82mpbird 247 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹))
841, 26, 57issubg2 17830 . . . 4 (𝐺 ∈ Grp → (ran 𝐹 ∈ (SubGrp‘𝐺) ↔ (ran 𝐹𝑋 ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹))))
8584adantr 472 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ↔ (ran 𝐹𝑋 ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹))))
869, 24, 83, 85mpbir3and 1428 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺))
8786, 22jca 555 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wss 3715  c0 4058  cmpt 4881  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  1c1 10149   + caddc 10151  -cneg 10479  cz 11589  Basecbs 16079  +gcplusg 16163  Grpcgrp 17643  invgcminusg 17644  .gcmg 17761  SubGrpcsubg 17809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-seq 13016  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-mulg 17762  df-subg 17812
This theorem is referenced by:  cycsubg  17843  oddvds2  18203  cycsubgcyg  18522
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