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Theorem dfod2 18152
 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 12937 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ∈ Fin)
2 odf1.1 . . . . . . . . . . . . 13 𝑋 = (Base‘𝐺)
3 odf1.3 . . . . . . . . . . . . 13 · = (.g𝐺)
42, 3mulgcl 17731 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
543expa 1111 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
65an32s 881 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
76adantlr 753 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
8 odf1.4 . . . . . . . . 9 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
97, 8fmptd 6536 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10 frn 6202 . . . . . . . 8 (𝐹:ℤ⟶𝑋 → ran 𝐹𝑋)
11 fvex 6350 . . . . . . . . . 10 (Base‘𝐺) ∈ V
122, 11eqeltri 2823 . . . . . . . . 9 𝑋 ∈ V
1312ssex 4942 . . . . . . . 8 (ran 𝐹𝑋 → ran 𝐹 ∈ V)
149, 10, 133syl 18 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ran 𝐹 ∈ V)
15 elfzelz 12506 . . . . . . . . . . 11 (𝑦 ∈ (0...((𝑂𝐴) − 1)) → 𝑦 ∈ ℤ)
1615adantl 473 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℤ)
17 ovex 6829 . . . . . . . . . 10 (𝑦 · 𝐴) ∈ V
18 oveq1 6808 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
198, 18elrnmpt1s 5516 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ (𝑦 · 𝐴) ∈ V) → (𝑦 · 𝐴) ∈ ran 𝐹)
2016, 17, 19sylancl 697 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 · 𝐴) ∈ ran 𝐹)
2120ralrimiva 3092 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
22 zmodfz 12857 . . . . . . . . . . . . 13 ((𝑥 ∈ ℤ ∧ (𝑂𝐴) ∈ ℕ) → (𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)))
2322ancoms 468 . . . . . . . . . . . 12 (((𝑂𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)))
2423adantll 752 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)))
25 simpllr 817 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑂𝐴) ∈ ℕ)
26 simplr 809 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑥 ∈ ℤ)
2715adantl 473 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℤ)
28 moddvds 15164 . . . . . . . . . . . . . 14 (((𝑂𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
2925, 26, 27, 28syl3anc 1463 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
3027zred 11645 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℝ)
3125nnrpd 12034 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑂𝐴) ∈ ℝ+)
32 0z 11551 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℤ
33 nnz 11562 . . . . . . . . . . . . . . . . . . . . 21 ((𝑂𝐴) ∈ ℕ → (𝑂𝐴) ∈ ℤ)
3433adantl 473 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) ∈ ℤ)
3534adantr 472 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑂𝐴) ∈ ℤ)
36 elfzm11 12575 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℤ ∧ (𝑂𝐴) ∈ ℤ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴))))
3732, 35, 36sylancr 698 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴))))
3837biimpa 502 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴)))
3938simp2d 1135 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 0 ≤ 𝑦)
4038simp3d 1136 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 < (𝑂𝐴))
41 modid 12860 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ (𝑂𝐴) ∈ ℝ+) ∧ (0 ≤ 𝑦𝑦 < (𝑂𝐴))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4230, 31, 39, 40, 41syl22anc 1464 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4342eqeq2d 2758 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑥 mod (𝑂𝐴)) = 𝑦))
44 eqcom 2755 . . . . . . . . . . . . . 14 ((𝑥 mod (𝑂𝐴)) = 𝑦𝑦 = (𝑥 mod (𝑂𝐴)))
4543, 44syl6bb 276 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
46 simp-4l 825 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝐺 ∈ Grp)
47 simp-4r 827 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝐴𝑋)
48 odf1.2 . . . . . . . . . . . . . . 15 𝑂 = (od‘𝐺)
49 eqid 2748 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
502, 48, 3, 49odcong 18139 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂𝐴) ∥ (𝑥𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5146, 47, 26, 27, 50syl112anc 1467 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑂𝐴) ∥ (𝑥𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5229, 45, 513bitr3rd 299 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
5352ralrimiva 3092 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
54 reu6i 3526 . . . . . . . . . . 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))
5524, 53, 54syl2anc 696 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
5655ralrimiva 3092 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
57 ovex 6829 . . . . . . . . . . 11 (𝑥 · 𝐴) ∈ V
5857rgenw 3050 . . . . . . . . . 10 𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
59 eqeq1 2752 . . . . . . . . . . . 12 (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
6059reubidv 3253 . . . . . . . . . . 11 (𝑧 = (𝑥 · 𝐴) → (∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴)))
618, 60ralrnmpt 6519 . . . . . . . . . 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)))
6258, 61ax-mp 5 . . . . . . . . 9 (∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
6356, 62sylibr 224 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴))
64 eqid 2748 . . . . . . . . 9 (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴))
6564f1ompt 6533 . . . . . . . 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)))
6621, 63, 65sylanbrc 701 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂𝐴) − 1))–1-1-onto→ran 𝐹)
67 f1oen2g 8126 . . . . . . 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)
681, 14, 66, 67syl3anc 1463 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ≈ ran 𝐹)
69 enfi 8329 . . . . . 6 ((0...((𝑂𝐴) − 1)) ≈ ran 𝐹 → ((0...((𝑂𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
7068, 69syl 17 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ((0...((𝑂𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
711, 70mpbid 222 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ran 𝐹 ∈ Fin)
7271iftrued 4226 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = (♯‘ran 𝐹))
73 fz01en 12533 . . . . . 6 ((𝑂𝐴) ∈ ℤ → (0...((𝑂𝐴) − 1)) ≈ (1...(𝑂𝐴)))
74 ensym 8158 . . . . . 6 ((0...((𝑂𝐴) − 1)) ≈ (1...(𝑂𝐴)) → (1...(𝑂𝐴)) ≈ (0...((𝑂𝐴) − 1)))
7534, 73, 743syl 18 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ≈ (0...((𝑂𝐴) − 1)))
76 entr 8161 . . . . 5 (((1...(𝑂𝐴)) ≈ (0...((𝑂𝐴) − 1)) ∧ (0...((𝑂𝐴) − 1)) ≈ ran 𝐹) → (1...(𝑂𝐴)) ≈ ran 𝐹)
7775, 68, 76syl2anc 696 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ≈ ran 𝐹)
78 fzfid 12937 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ∈ Fin)
79 hashen 13300 . . . . 5 (((1...(𝑂𝐴)) ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((♯‘(1...(𝑂𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂𝐴)) ≈ ran 𝐹))
8078, 71, 79syl2anc 696 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ((♯‘(1...(𝑂𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂𝐴)) ≈ ran 𝐹))
8177, 80mpbird 247 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (♯‘(1...(𝑂𝐴))) = (♯‘ran 𝐹))
82 nnnn0 11462 . . . . 5 ((𝑂𝐴) ∈ ℕ → (𝑂𝐴) ∈ ℕ0)
8382adantl 473 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) ∈ ℕ0)
84 hashfz1 13299 . . . 4 ((𝑂𝐴) ∈ ℕ0 → (♯‘(1...(𝑂𝐴))) = (𝑂𝐴))
8583, 84syl 17 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (♯‘(1...(𝑂𝐴))) = (𝑂𝐴))
8672, 81, 853eqtr2rd 2789 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
87 simp3 1130 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = 0)
882, 48, 3, 8odinf 18151 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
8988iffalsed 4229 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = 0)
9087, 89eqtr4d 2785 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
91903expa 1111 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
922, 48odcl 18126 . . . 4 (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
9392adantl 473 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ0)
94 elnn0 11457 . . 3 ((𝑂𝐴) ∈ ℕ0 ↔ ((𝑂𝐴) ∈ ℕ ∨ (𝑂𝐴) = 0))
9593, 94sylib 208 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) ∈ ℕ ∨ (𝑂𝐴) = 0))
9686, 91, 95mpjaodan 862 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1620   ∈ wcel 2127  ∀wral 3038  ∃!wreu 3040  Vcvv 3328   ⊆ wss 3703  ifcif 4218   class class class wbr 4792   ↦ cmpt 4869  ran crn 5255  ⟶wf 6033  –1-1-onto→wf1o 6036  ‘cfv 6037  (class class class)co 6801   ≈ cen 8106  Fincfn 8109  ℝcr 10098  0cc0 10099  1c1 10100   < clt 10237   ≤ cle 10238   − cmin 10429  ℕcn 11183  ℕ0cn0 11455  ℤcz 11540  ℝ+crp 11996  ...cfz 12490   mod cmo 12833  ♯chash 13282   ∥ cdvds 15153  Basecbs 16030  0gc0g 16273  Grpcgrp 17594  .gcmg 17712  odcod 18115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-inf2 8699  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-se 5214  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-isom 6046  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-omul 7722  df-er 7899  df-map 8013  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8501  df-inf 8502  df-oi 8568  df-card 8926  df-acn 8929  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-div 10848  df-nn 11184  df-2 11242  df-3 11243  df-n0 11456  df-z 11541  df-uz 11851  df-rp 11997  df-fz 12491  df-fl 12758  df-mod 12834  df-seq 12967  df-exp 13026  df-hash 13283  df-cj 14009  df-re 14010  df-im 14011  df-sqrt 14145  df-abs 14146  df-dvds 15154  df-0g 16275  df-mgm 17414  df-sgrp 17456  df-mnd 17467  df-grp 17597  df-minusg 17598  df-sbg 17599  df-mulg 17713  df-od 18119 This theorem is referenced by:  oddvds2  18154  cyggenod  18457  cyggenod2  18458
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