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Theorem cycsubgcl 17443
 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 17382 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
433expa 1257 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
54an32s 842 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
6 cycsubg.f . . . . 5 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
75, 6fmptd 6292 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:ℤ⟶𝑋)
8 frn 5966 . . . 4 (𝐹:ℤ⟶𝑋 → ran 𝐹𝑋)
97, 8syl 17 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹𝑋)
10 1z 11284 . . . . . . 7 1 ∈ ℤ
11 oveq1 6556 . . . . . . . 8 (𝑥 = 1 → (𝑥 · 𝐴) = (1 · 𝐴))
12 ovex 6577 . . . . . . . 8 (1 · 𝐴) ∈ V
1311, 6, 12fvmpt 6191 . . . . . . 7 (1 ∈ ℤ → (𝐹‘1) = (1 · 𝐴))
1410, 13ax-mp 5 . . . . . 6 (𝐹‘1) = (1 · 𝐴)
151, 2mulg1 17371 . . . . . . 7 (𝐴𝑋 → (1 · 𝐴) = 𝐴)
1615adantl 481 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (1 · 𝐴) = 𝐴)
1714, 16syl5eq 2656 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) = 𝐴)
18 ffn 5958 . . . . . . 7 (𝐹:ℤ⟶𝑋𝐹 Fn ℤ)
197, 18syl 17 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹 Fn ℤ)
20 fnfvelrn 6264 . . . . . 6 ((𝐹 Fn ℤ ∧ 1 ∈ ℤ) → (𝐹‘1) ∈ ran 𝐹)
2119, 10, 20sylancl 693 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) ∈ ran 𝐹)
2217, 21eqeltrrd 2689 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐴 ∈ ran 𝐹)
23 ne0i 3880 . . . 4 (𝐴 ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
2422, 23syl 17 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ≠ ∅)
25 df-3an 1033 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋))
26 eqid 2610 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
271, 2, 26mulgdir 17396 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
2825, 27sylan2br 492 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
2928anass1rs 845 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
30 zaddcl 11294 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 + 𝑛) ∈ ℤ)
3130adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) ∈ ℤ)
32 oveq1 6556 . . . . . . . . . . . . 13 (𝑥 = (𝑚 + 𝑛) → (𝑥 · 𝐴) = ((𝑚 + 𝑛) · 𝐴))
33 ovex 6577 . . . . . . . . . . . . 13 ((𝑚 + 𝑛) · 𝐴) ∈ V
3432, 6, 33fvmpt 6191 . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . 14 (𝑥 = 𝑚 → (𝑥 · 𝐴) = (𝑚 · 𝐴))
37 ovex 6577 . . . . . . . . . . . . . 14 (𝑚 · 𝐴) ∈ V
3836, 6, 37fvmpt 6191 . . . . . . . . . . . . 13 (𝑚 ∈ ℤ → (𝐹𝑚) = (𝑚 · 𝐴))
3938ad2antrl 760 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑚) = (𝑚 · 𝐴))
40 oveq1 6556 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (𝑥 · 𝐴) = (𝑛 · 𝐴))
41 ovex 6577 . . . . . . . . . . . . . 14 (𝑛 · 𝐴) ∈ V
4240, 6, 41fvmpt 6191 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (𝐹𝑛) = (𝑛 · 𝐴))
4342ad2antll 761 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑛) = (𝑛 · 𝐴))
4439, 43oveq12d 6567 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
4529, 35, 443eqtr4d 2654 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
46 fnfvelrn 6264 . . . . . . . . . . 11 ((𝐹 Fn ℤ ∧ (𝑚 + 𝑛) ∈ ℤ) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
4719, 30, 46syl2an 493 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
4845, 47eqeltrrd 2689 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
4948anassrs 678 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) ∧ 𝑛 ∈ ℤ) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
5049ralrimiva 2949 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
51 oveq2 6557 . . . . . . . . . . 11 (𝑣 = (𝐹𝑛) → ((𝐹𝑚)(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
5251eleq1d 2672 . . . . . . . . . 10 (𝑣 = (𝐹𝑛) → (((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
5352ralrn 6270 . . . . . . . . 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 480 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
5650, 55mpbird 246 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹)
57 eqid 2610 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
581, 2, 57mulgneg 17383 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
59583expa 1257 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ) ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
6059an32s 842 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
61 znegcl 11289 . . . . . . . . . 10 (𝑚 ∈ ℤ → -𝑚 ∈ ℤ)
6261adantl 481 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → -𝑚 ∈ ℤ)
63 oveq1 6556 . . . . . . . . . 10 (𝑥 = -𝑚 → (𝑥 · 𝐴) = (-𝑚 · 𝐴))
64 ovex 6577 . . . . . . . . . 10 (-𝑚 · 𝐴) ∈ V
6563, 6, 64fvmpt 6191 . . . . . . . . 9 (-𝑚 ∈ ℤ → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
6662, 65syl 17 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
6738adantl 481 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹𝑚) = (𝑚 · 𝐴))
6867fveq2d 6107 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) = ((invg𝐺)‘(𝑚 · 𝐴)))
6960, 66, 683eqtr4d 2654 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = ((invg𝐺)‘(𝐹𝑚)))
70 fnfvelrn 6264 . . . . . . . 8 ((𝐹 Fn ℤ ∧ -𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
7119, 61, 70syl2an 493 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
7269, 71eqeltrrd 2689 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹)
7356, 72jca 553 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
7473ralrimiva 2949 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
75 oveq1 6556 . . . . . . . . 9 (𝑢 = (𝐹𝑚) → (𝑢(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)𝑣))
7675eleq1d 2672 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
7776ralbidv 2969 . . . . . . 7 (𝑢 = (𝐹𝑚) → (∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
78 fveq2 6103 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((invg𝐺)‘𝑢) = ((invg𝐺)‘(𝐹𝑚)))
7978eleq1d 2672 . . . . . . 7 (𝑢 = (𝐹𝑚) → (((invg𝐺)‘𝑢) ∈ ran 𝐹 ↔ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
8077, 79anbi12d 743 . . . . . 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 6270 . . . . 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 246 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹))
841, 26, 57issubg2 17432 . . . 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 480 . . 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 1238 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺))
8786, 22jca 553 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  -cneg 10146  ℤcz 11254  Basecbs 15695  +gcplusg 15768  Grpcgrp 17245  invgcminusg 17246  .gcmg 17363  SubGrpcsubg 17411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mulg 17364  df-subg 17414 This theorem is referenced by:  cycsubg  17445  oddvds2  17806  cycsubgcyg  18125
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