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Theorem cycsubgcl 18431
 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 18327 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
433expa 1116 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
54an32s 651 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
6 cycsubg.f . . . . 5 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
75, 6fmptd 6876 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹:ℤ⟶𝑋)
87frnd 6511 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹𝑋)
97ffnd 6505 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐹 Fn ℤ)
10 1z 12065 . . . . 5 1 ∈ ℤ
11 fnfvelrn 6846 . . . . 5 ((𝐹 Fn ℤ ∧ 1 ∈ ℤ) → (𝐹‘1) ∈ ran 𝐹)
129, 10, 11sylancl 589 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) ∈ ran 𝐹)
1312ne0d 4237 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ≠ ∅)
14 df-3an 1087 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋))
15 eqid 2759 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
161, 2, 15mulgdir 18341 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
1714, 16sylan2br 597 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
1817anass1rs 654 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
19 zaddcl 12075 . . . . . . . . . . . . 13 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 + 𝑛) ∈ ℤ)
2019adantl 485 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) ∈ ℤ)
21 oveq1 7164 . . . . . . . . . . . . 13 (𝑥 = (𝑚 + 𝑛) → (𝑥 · 𝐴) = ((𝑚 + 𝑛) · 𝐴))
22 ovex 7190 . . . . . . . . . . . . 13 ((𝑚 + 𝑛) · 𝐴) ∈ V
2321, 6, 22fvmpt 6765 . . . . . . . . . . . 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 7164 . . . . . . . . . . . . . 14 (𝑥 = 𝑚 → (𝑥 · 𝐴) = (𝑚 · 𝐴))
26 ovex 7190 . . . . . . . . . . . . . 14 (𝑚 · 𝐴) ∈ V
2725, 6, 26fvmpt 6765 . . . . . . . . . . . . 13 (𝑚 ∈ ℤ → (𝐹𝑚) = (𝑚 · 𝐴))
2827ad2antrl 727 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑚) = (𝑚 · 𝐴))
29 oveq1 7164 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (𝑥 · 𝐴) = (𝑛 · 𝐴))
30 ovex 7190 . . . . . . . . . . . . . 14 (𝑛 · 𝐴) ∈ V
3129, 6, 30fvmpt 6765 . . . . . . . . . . . . 13 (𝑛 ∈ ℤ → (𝐹𝑛) = (𝑛 · 𝐴))
3231ad2antll 728 . . . . . . . . . . . 12 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹𝑛) = (𝑛 · 𝐴))
3328, 32oveq12d 7175 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) = ((𝑚 · 𝐴)(+g𝐺)(𝑛 · 𝐴)))
3418, 24, 333eqtr4d 2804 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
35 fnfvelrn 6846 . . . . . . . . . . 11 ((𝐹 Fn ℤ ∧ (𝑚 + 𝑛) ∈ ℤ) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
369, 19, 35syl2an 598 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
3734, 36eqeltrrd 2854 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
3837anassrs 471 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) ∧ 𝑛 ∈ ℤ) → ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
3938ralrimiva 3114 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹)
40 oveq2 7165 . . . . . . . . . . 11 (𝑣 = (𝐹𝑛) → ((𝐹𝑚)(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)(𝐹𝑛)))
4140eleq1d 2837 . . . . . . . . . 10 (𝑣 = (𝐹𝑛) → (((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
4241ralrn 6852 . . . . . . . . 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 484 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹𝑚)(+g𝐺)(𝐹𝑛)) ∈ ran 𝐹))
4539, 44mpbird 260 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹)
46 eqid 2759 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
471, 2, 46mulgneg 18328 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
48473expa 1116 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ) ∧ 𝐴𝑋) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
4948an32s 651 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (-𝑚 · 𝐴) = ((invg𝐺)‘(𝑚 · 𝐴)))
50 znegcl 12070 . . . . . . . . . 10 (𝑚 ∈ ℤ → -𝑚 ∈ ℤ)
5150adantl 485 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → -𝑚 ∈ ℤ)
52 oveq1 7164 . . . . . . . . . 10 (𝑥 = -𝑚 → (𝑥 · 𝐴) = (-𝑚 · 𝐴))
53 ovex 7190 . . . . . . . . . 10 (-𝑚 · 𝐴) ∈ V
5452, 6, 53fvmpt 6765 . . . . . . . . 9 (-𝑚 ∈ ℤ → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
5551, 54syl 17 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
5627adantl 485 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹𝑚) = (𝑚 · 𝐴))
5756fveq2d 6668 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) = ((invg𝐺)‘(𝑚 · 𝐴)))
5849, 55, 573eqtr4d 2804 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = ((invg𝐺)‘(𝐹𝑚)))
59 fnfvelrn 6846 . . . . . . . 8 ((𝐹 Fn ℤ ∧ -𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
609, 50, 59syl2an 598 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
6158, 60eqeltrrd 2854 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹)
6245, 61jca 515 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
6362ralrimiva 3114 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
64 oveq1 7164 . . . . . . . . 9 (𝑢 = (𝐹𝑚) → (𝑢(+g𝐺)𝑣) = ((𝐹𝑚)(+g𝐺)𝑣))
6564eleq1d 2837 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
6665ralbidv 3127 . . . . . . 7 (𝑢 = (𝐹𝑚) → (∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑣 ∈ ran 𝐹((𝐹𝑚)(+g𝐺)𝑣) ∈ ran 𝐹))
67 fveq2 6664 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((invg𝐺)‘𝑢) = ((invg𝐺)‘(𝐹𝑚)))
6867eleq1d 2837 . . . . . . 7 (𝑢 = (𝐹𝑚) → (((invg𝐺)‘𝑢) ∈ ran 𝐹 ↔ ((invg𝐺)‘(𝐹𝑚)) ∈ ran 𝐹))
6966, 68anbi12d 633 . . . . . 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 6852 . . . . 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 260 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg𝐺)‘𝑢) ∈ ran 𝐹))
731, 15, 46issubg2 18376 . . . 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 484 . . 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 1340 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺))
76 oveq1 7164 . . . . . 6 (𝑥 = 1 → (𝑥 · 𝐴) = (1 · 𝐴))
77 ovex 7190 . . . . . 6 (1 · 𝐴) ∈ V
7876, 6, 77fvmpt 6765 . . . . 5 (1 ∈ ℤ → (𝐹‘1) = (1 · 𝐴))
7910, 78ax-mp 5 . . . 4 (𝐹‘1) = (1 · 𝐴)
801, 2mulg1 18317 . . . . 5 (𝐴𝑋 → (1 · 𝐴) = 𝐴)
8180adantl 485 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (1 · 𝐴) = 𝐴)
8279, 81syl5eq 2806 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹‘1) = 𝐴)
8382, 12eqeltrrd 2854 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → 𝐴 ∈ ran 𝐹)
8475, 83jca 515 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071   ⊆ wss 3861  ∅c0 4228   ↦ cmpt 5117  ran crn 5530   Fn wfn 6336  ‘cfv 6341  (class class class)co 7157  1c1 10590   + caddc 10592  -cneg 10923  ℤcz 12034  Basecbs 16556  +gcplusg 16638  Grpcgrp 18184  invgcminusg 18185  .gcmg 18306  SubGrpcsubg 18355 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-cnex 10645  ax-resscn 10646  ax-1cn 10647  ax-icn 10648  ax-addcl 10649  ax-addrcl 10650  ax-mulcl 10651  ax-mulrcl 10652  ax-mulcom 10653  ax-addass 10654  ax-mulass 10655  ax-distr 10656  ax-i2m1 10657  ax-1ne0 10658  ax-1rid 10659  ax-rnegex 10660  ax-rrecex 10661  ax-cnre 10662  ax-pre-lttri 10663  ax-pre-lttrn 10664  ax-pre-ltadd 10665  ax-pre-mulgt0 10666 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-er 8306  df-en 8542  df-dom 8543  df-sdom 8544  df-pnf 10729  df-mnf 10730  df-xr 10731  df-ltxr 10732  df-le 10733  df-sub 10924  df-neg 10925  df-nn 11689  df-2 11751  df-n0 11949  df-z 12035  df-uz 12297  df-fz 12954  df-seq 13433  df-ndx 16559  df-slot 16560  df-base 16562  df-sets 16563  df-ress 16564  df-plusg 16651  df-0g 16788  df-mgm 17933  df-sgrp 17982  df-mnd 17993  df-grp 18187  df-minusg 18188  df-mulg 18307  df-subg 18358 This theorem is referenced by:  cycsubg  18433  cycsubgcld  18434  oddvds2  18775  cycsubgcyg  19104
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