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Theorem dfod2 18691
Description: An alternative definition of the order of a group element is as the cardinality of the cyclic subgroup generated by the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
odf1.1 𝑋 = (Base‘𝐺)
odf1.2 𝑂 = (od‘𝐺)
odf1.3 · = (.g𝐺)
odf1.4 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
Assertion
Ref Expression
dfod2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥,𝑂   𝑥, ·   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem dfod2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 13345 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ∈ Fin)
2 odf1.1 . . . . . . . . . . . . 13 𝑋 = (Base‘𝐺)
3 odf1.3 . . . . . . . . . . . . 13 · = (.g𝐺)
42, 3mulgcl 18245 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
543expa 1115 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
65an32s 651 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
76adantlr 714 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
8 odf1.4 . . . . . . . . 9 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
97, 8fmptd 6869 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10 frn 6509 . . . . . . . 8 (𝐹:ℤ⟶𝑋 → ran 𝐹𝑋)
112fvexi 6675 . . . . . . . . 9 𝑋 ∈ V
1211ssex 5211 . . . . . . . 8 (ran 𝐹𝑋 → ran 𝐹 ∈ V)
139, 10, 123syl 18 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ran 𝐹 ∈ V)
14 elfzelz 12911 . . . . . . . . . . 11 (𝑦 ∈ (0...((𝑂𝐴) − 1)) → 𝑦 ∈ ℤ)
1514adantl 485 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℤ)
16 ovex 7182 . . . . . . . . . 10 (𝑦 · 𝐴) ∈ V
17 oveq1 7156 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
188, 17elrnmpt1s 5816 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ (𝑦 · 𝐴) ∈ V) → (𝑦 · 𝐴) ∈ ran 𝐹)
1915, 16, 18sylancl 589 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 · 𝐴) ∈ ran 𝐹)
2019ralrimiva 3177 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
21 zmodfz 13265 . . . . . . . . . . . . 13 ((𝑥 ∈ ℤ ∧ (𝑂𝐴) ∈ ℕ) → (𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)))
2221ancoms 462 . . . . . . . . . . . 12 (((𝑂𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)))
2322adantll 713 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)))
24 simpllr 775 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑂𝐴) ∈ ℕ)
25 simplr 768 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑥 ∈ ℤ)
2614adantl 485 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℤ)
27 moddvds 15618 . . . . . . . . . . . . . 14 (((𝑂𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
2824, 25, 26, 27syl3anc 1368 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
2926zred 12084 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℝ)
3024nnrpd 12426 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑂𝐴) ∈ ℝ+)
31 0z 11989 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℤ
32 nnz 12001 . . . . . . . . . . . . . . . . . . . . 21 ((𝑂𝐴) ∈ ℕ → (𝑂𝐴) ∈ ℤ)
3332adantl 485 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) ∈ ℤ)
3433adantr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑂𝐴) ∈ ℤ)
35 elfzm11 12982 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℤ ∧ (𝑂𝐴) ∈ ℤ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴))))
3631, 34, 35sylancr 590 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴))))
3736biimpa 480 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴)))
3837simp2d 1140 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 0 ≤ 𝑦)
3937simp3d 1141 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 < (𝑂𝐴))
40 modid 13268 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ (𝑂𝐴) ∈ ℝ+) ∧ (0 ≤ 𝑦𝑦 < (𝑂𝐴))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4129, 30, 38, 39, 40syl22anc 837 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4241eqeq2d 2835 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑥 mod (𝑂𝐴)) = 𝑦))
43 eqcom 2831 . . . . . . . . . . . . . 14 ((𝑥 mod (𝑂𝐴)) = 𝑦𝑦 = (𝑥 mod (𝑂𝐴)))
4442, 43syl6bb 290 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
45 simp-4l 782 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝐺 ∈ Grp)
46 simp-4r 783 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝐴𝑋)
47 odf1.2 . . . . . . . . . . . . . . 15 𝑂 = (od‘𝐺)
48 eqid 2824 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
492, 47, 3, 48odcong 18677 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂𝐴) ∥ (𝑥𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5045, 46, 25, 26, 49syl112anc 1371 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑂𝐴) ∥ (𝑥𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5128, 44, 503bitr3rd 313 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
5251ralrimiva 3177 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
53 reu6i 3705 . . . . . . . . . . 11 (((𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)) ∧ ∀𝑦 ∈ (0...((𝑂𝐴) − 1))((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴)))) → ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
5423, 52, 53syl2anc 587 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
5554ralrimiva 3177 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
56 ovex 7182 . . . . . . . . . . 11 (𝑥 · 𝐴) ∈ V
5756rgenw 3145 . . . . . . . . . 10 𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
58 eqeq1 2828 . . . . . . . . . . . 12 (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5958reubidv 3380 . . . . . . . . . . 11 (𝑧 = (𝑥 · 𝐴) → (∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴)))
608, 59ralrnmptw 6851 . . . . . . . . . 10 (∀𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V → (∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴)))
6157, 60ax-mp 5 . . . . . . . . 9 (∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
6255, 61sylibr 237 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴))
63 eqid 2824 . . . . . . . . 9 (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴))
6463f1ompt 6866 . . . . . . . 8 ((𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂𝐴) − 1))–1-1-onto→ran 𝐹 ↔ (∀𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹 ∧ ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴)))
6520, 62, 64sylanbrc 586 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂𝐴) − 1))–1-1-onto→ran 𝐹)
66 f1oen2g 8522 . . . . . . 7 (((0...((𝑂𝐴) − 1)) ∈ Fin ∧ ran 𝐹 ∈ V ∧ (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂𝐴) − 1))–1-1-onto→ran 𝐹) → (0...((𝑂𝐴) − 1)) ≈ ran 𝐹)
671, 13, 65, 66syl3anc 1368 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ≈ ran 𝐹)
68 enfi 8731 . . . . . 6 ((0...((𝑂𝐴) − 1)) ≈ ran 𝐹 → ((0...((𝑂𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
6967, 68syl 17 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ((0...((𝑂𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
701, 69mpbid 235 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ran 𝐹 ∈ Fin)
7170iftrued 4458 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = (♯‘ran 𝐹))
72 fz01en 12939 . . . . . 6 ((𝑂𝐴) ∈ ℤ → (0...((𝑂𝐴) − 1)) ≈ (1...(𝑂𝐴)))
73 ensym 8554 . . . . . 6 ((0...((𝑂𝐴) − 1)) ≈ (1...(𝑂𝐴)) → (1...(𝑂𝐴)) ≈ (0...((𝑂𝐴) − 1)))
7433, 72, 733syl 18 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ≈ (0...((𝑂𝐴) − 1)))
75 entr 8557 . . . . 5 (((1...(𝑂𝐴)) ≈ (0...((𝑂𝐴) − 1)) ∧ (0...((𝑂𝐴) − 1)) ≈ ran 𝐹) → (1...(𝑂𝐴)) ≈ ran 𝐹)
7674, 67, 75syl2anc 587 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ≈ ran 𝐹)
77 fzfid 13345 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ∈ Fin)
78 hashen 13712 . . . . 5 (((1...(𝑂𝐴)) ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((♯‘(1...(𝑂𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂𝐴)) ≈ ran 𝐹))
7977, 70, 78syl2anc 587 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ((♯‘(1...(𝑂𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂𝐴)) ≈ ran 𝐹))
8076, 79mpbird 260 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (♯‘(1...(𝑂𝐴))) = (♯‘ran 𝐹))
81 nnnn0 11901 . . . . 5 ((𝑂𝐴) ∈ ℕ → (𝑂𝐴) ∈ ℕ0)
8281adantl 485 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) ∈ ℕ0)
83 hashfz1 13711 . . . 4 ((𝑂𝐴) ∈ ℕ0 → (♯‘(1...(𝑂𝐴))) = (𝑂𝐴))
8482, 83syl 17 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (♯‘(1...(𝑂𝐴))) = (𝑂𝐴))
8571, 80, 843eqtr2rd 2866 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
86 simp3 1135 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = 0)
872, 47, 3, 8odinf 18690 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
8887iffalsed 4461 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = 0)
8986, 88eqtr4d 2862 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
90893expa 1115 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
912, 47odcl 18664 . . . 4 (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
9291adantl 485 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ0)
93 elnn0 11896 . . 3 ((𝑂𝐴) ∈ ℕ0 ↔ ((𝑂𝐴) ∈ ℕ ∨ (𝑂𝐴) = 0))
9492, 93sylib 221 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) ∈ ℕ ∨ (𝑂𝐴) = 0))
9585, 90, 94mpjaodan 956 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wral 3133  ∃!wreu 3135  Vcvv 3480  wss 3919  ifcif 4450   class class class wbr 5052  cmpt 5132  ran crn 5543  wf 6339  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  cen 8502  Fincfn 8505  cr 10534  0cc0 10535  1c1 10536   < clt 10673  cle 10674  cmin 10868  cn 11634  0cn0 11894  cz 11978  +crp 12386  ...cfz 12894   mod cmo 13241  chash 13695  cdvds 15607  Basecbs 16483  0gc0g 16713  Grpcgrp 18103  .gcmg 18224  odcod 18652
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-omul 8103  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-card 9365  df-acn 9368  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-fz 12895  df-fl 13166  df-mod 13242  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-dvds 15608  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-od 18656
This theorem is referenced by:  oddvds2  18693  cyggenod  19003  cyggenod2  19004  cycsubggenodd  19231
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