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Theorem cycsubgcl 19049
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 18943 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
433expa 1118 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
54an32s 650 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
6 cycsubg.f . . . . 5 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
75, 6fmptd 7098 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:ℤ⟶𝑋)
87frnd 6712 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹𝑋)
97ffnd 6705 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹 Fn ℤ)
10 1z 12574 . . . . 5 1 ∈ ℤ
11 fnfvelrn 7067 . . . . 5 ((𝐹 Fn ℤ ∧ 1 ∈ ℤ) → (𝐹‘1) ∈ ran 𝐹)
129, 10, 11sylancl 586 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) ∈ ran 𝐹)
1312ne0d 4331 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ≠ ∅)
14 df-3an 1089 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋))
15 eqid 2731 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
161, 2, 15mulgdir 18958 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
1714, 16sylan2br 595 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
1817anass1rs 653 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
19 zaddcl 12584 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 + 𝑛) ∈ ℤ)
2019adantl 482 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) ∈ ℤ)
21 oveq1 7400 . . . . . . . . . . . . 13 (𝑥 = (𝑚 + 𝑛) → (𝑥 · 𝐴) = ((𝑚 + 𝑛) · 𝐴))
22 ovex 7426 . . . . . . . . . . . . 13 ((𝑚 + 𝑛) · 𝐴) ∈ V
2321, 6, 22fvmpt 6984 . . . . . . . . . . . 12 ((𝑚 + 𝑛) ∈ ℤ → (𝐹‘(𝑚 + 𝑛)) = ((𝑚 + 𝑛) · 𝐴))
2420, 23syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝑚 + 𝑛) · 𝐴))
25 oveq1 7400 . . . . . . . . . . . . . 14 (𝑥 = 𝑚 → (𝑥 · 𝐴) = (𝑚 · 𝐴))
26 ovex 7426 . . . . . . . . . . . . . 14 (𝑚 · 𝐴) ∈ V
2725, 6, 26fvmpt 6984 . . . . . . . . . . . . 13 (𝑚 ∈ ℤ → (𝐹𝑚) = (𝑚 · 𝐴))
2827ad2antrl 726 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑚) = (𝑚 · 𝐴))
29 oveq1 7400 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (𝑥 · 𝐴) = (𝑛 · 𝐴))
30 ovex 7426 . . . . . . . . . . . . . 14 (𝑛 · 𝐴) ∈ V
3129, 6, 30fvmpt 6984 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (𝐹𝑛) = (𝑛 · 𝐴))
3231ad2antll 727 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑛) = (𝑛 · 𝐴))
3328, 32oveq12d 7411 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
3418, 24, 333eqtr4d 2781 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
35 fnfvelrn 7067 . . . . . . . . . . 11 ((𝐹 Fn ℤ ∧ (𝑚 + 𝑛) ∈ ℤ) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
369, 19, 35syl2an 596 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
3734, 36eqeltrrd 2833 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
3837anassrs 468 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) ∧ 𝑛 ∈ ℤ) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
3938ralrimiva 3145 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
40 oveq2 7401 . . . . . . . . . . 11 (𝑣 = (𝐹𝑛) → ((𝐹𝑚)(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
4140eleq1d 2817 . . . . . . . . . 10 (𝑣 = (𝐹𝑛) → (((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
4241ralrn 7074 . . . . . . . . 9 (𝐹 Fn ℤ → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
439, 42syl 17 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
4443adantr 481 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
4539, 44mpbird 256 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹)
46 eqid 2731 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
471, 2, 46mulgneg 18944 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
48473expa 1118 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ) ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
4948an32s 650 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
50 znegcl 12579 . . . . . . . . . 10 (𝑚 ∈ ℤ → -𝑚 ∈ ℤ)
5150adantl 482 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → -𝑚 ∈ ℤ)
52 oveq1 7400 . . . . . . . . . 10 (𝑥 = -𝑚 → (𝑥 · 𝐴) = (-𝑚 · 𝐴))
53 ovex 7426 . . . . . . . . . 10 (-𝑚 · 𝐴) ∈ V
5452, 6, 53fvmpt 6984 . . . . . . . . 9 (-𝑚 ∈ ℤ → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
5551, 54syl 17 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
5627adantl 482 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹𝑚) = (𝑚 · 𝐴))
5756fveq2d 6882 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) = ((invg𝐺)‘(𝑚 · 𝐴)))
5849, 55, 573eqtr4d 2781 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = ((invg𝐺)‘(𝐹𝑚)))
59 fnfvelrn 7067 . . . . . . . 8 ((𝐹 Fn ℤ ∧ -𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
609, 50, 59syl2an 596 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
6158, 60eqeltrrd 2833 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹)
6245, 61jca 512 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
6362ralrimiva 3145 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
64 oveq1 7400 . . . . . . . . 9 (𝑢 = (𝐹𝑚) → (𝑢(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)𝑣))
6564eleq1d 2817 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
6665ralbidv 3176 . . . . . . 7 (𝑢 = (𝐹𝑚) → (∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
67 fveq2 6878 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((invg𝐺)‘𝑢) = ((invg𝐺)‘(𝐹𝑚)))
6867eleq1d 2817 . . . . . . 7 (𝑢 = (𝐹𝑚) → (((invg𝐺)‘𝑢) ∈ ran 𝐹 ↔ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
6966, 68anbi12d 631 . . . . . 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)))
7069ralrn 7074 . . . . 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)))
719, 70syl 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)))
7263, 71mpbird 256 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹))
731, 15, 46issubg2 18993 . . . 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))))
7473adantr 481 . . 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))))
758, 13, 72, 74mpbir3and 1342 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺))
76 oveq1 7400 . . . . . 6 (𝑥 = 1 → (𝑥 · 𝐴) = (1 · 𝐴))
77 ovex 7426 . . . . . 6 (1 · 𝐴) ∈ V
7876, 6, 77fvmpt 6984 . . . . 5 (1 ∈ ℤ → (𝐹‘1) = (1 · 𝐴))
7910, 78ax-mp 5 . . . 4 (𝐹‘1) = (1 · 𝐴)
801, 2mulg1 18933 . . . . 5 (𝐴𝑋 → (1 · 𝐴) = 𝐴)
8180adantl 482 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (1 · 𝐴) = 𝐴)
8279, 81eqtrid 2783 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) = 𝐴)
8382, 12eqeltrrd 2833 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐴 ∈ ran 𝐹)
8475, 83jca 512 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  wral 3060  wss 3944  c0 4318  cmpt 5224  ran crn 5670   Fn wfn 6527  cfv 6532  (class class class)co 7393  1c1 11093   + caddc 11095  -cneg 11427  cz 12540  Basecbs 17126  +gcplusg 17179  Grpcgrp 18794  invgcminusg 18795  .gcmg 18922  SubGrpcsubg 18972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-seq 13949  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-minusg 18798  df-mulg 18923  df-subg 18975
This theorem is referenced by:  cycsubg  19051  cycsubgcld  19052  oddvds2  19398  cycsubgcyg  19728
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