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Theorem dfod2 18179
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 12964 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ∈ Fin)
2 odf1.1 . . . . . . . . . . . . 13 𝑋 = (Base‘𝐺)
3 odf1.3 . . . . . . . . . . . . 13 · = (.g𝐺)
42, 3mulgcl 17758 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
543expa 1112 . . . . . . . . . . 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 6546 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10 frn 6212 . . . . . . . 8 (𝐹:ℤ⟶𝑋 → ran 𝐹𝑋)
11 fvex 6360 . . . . . . . . . 10 (Base‘𝐺) ∈ V
122, 11eqeltri 2833 . . . . . . . . 9 𝑋 ∈ V
1312ssex 4952 . . . . . . . 8 (ran 𝐹𝑋 → ran 𝐹 ∈ V)
149, 10, 133syl 18 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ran 𝐹 ∈ V)
15 elfzelz 12533 . . . . . . . . . . 11 (𝑦 ∈ (0...((𝑂𝐴) − 1)) → 𝑦 ∈ ℤ)
1615adantl 473 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℤ)
17 ovex 6839 . . . . . . . . . 10 (𝑦 · 𝐴) ∈ V
18 oveq1 6818 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
198, 18elrnmpt1s 5526 . . . . . . . . . 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 3102 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
22 zmodfz 12884 . . . . . . . . . . . . 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 15191 . . . . . . . . . . . . . 14 (((𝑂𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
2925, 26, 27, 28syl3anc 1477 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
3027zred 11672 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℝ)
3125nnrpd 12061 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑂𝐴) ∈ ℝ+)
32 0z 11578 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℤ
33 nnz 11589 . . . . . . . . . . . . . . . . . . . . 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 12602 . . . . . . . . . . . . . . . . . . 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 1138 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 0 ≤ 𝑦)
4038simp3d 1139 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 < (𝑂𝐴))
41 modid 12887 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ (𝑂𝐴) ∈ ℝ+) ∧ (0 ≤ 𝑦𝑦 < (𝑂𝐴))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4230, 31, 39, 40, 41syl22anc 1478 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4342eqeq2d 2768 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑥 mod (𝑂𝐴)) = 𝑦))
44 eqcom 2765 . . . . . . . . . . . . . 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 2758 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
502, 48, 3, 49odcong 18166 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂𝐴) ∥ (𝑥𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5146, 47, 26, 27, 50syl112anc 1481 . . . . . . . . . . . . 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 3102 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
54 reu6i 3536 . . . . . . . . . . 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 3102 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
57 ovex 6839 . . . . . . . . . . 11 (𝑥 · 𝐴) ∈ V
5857rgenw 3060 . . . . . . . . . 10 𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
59 eqeq1 2762 . . . . . . . . . . . 12 (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
6059reubidv 3263 . . . . . . . . . . 11 (𝑧 = (𝑥 · 𝐴) → (∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴)))
618, 60ralrnmpt 6529 . . . . . . . . . 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 2758 . . . . . . . . 9 (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴))
6564f1ompt 6543 . . . . . . . 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 8136 . . . . . . 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 1477 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ≈ ran 𝐹)
69 enfi 8339 . . . . . 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 4236 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = (♯‘ran 𝐹))
73 fz01en 12560 . . . . . 6 ((𝑂𝐴) ∈ ℤ → (0...((𝑂𝐴) − 1)) ≈ (1...(𝑂𝐴)))
74 ensym 8168 . . . . . 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 8171 . . . . 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 12964 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ∈ Fin)
79 hashen 13327 . . . . 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 11489 . . . . 5 ((𝑂𝐴) ∈ ℕ → (𝑂𝐴) ∈ ℕ0)
8382adantl 473 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) ∈ ℕ0)
84 hashfz1 13326 . . . 4 ((𝑂𝐴) ∈ ℕ0 → (♯‘(1...(𝑂𝐴))) = (𝑂𝐴))
8583, 84syl 17 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (♯‘(1...(𝑂𝐴))) = (𝑂𝐴))
8672, 81, 853eqtr2rd 2799 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
87 simp3 1133 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = 0)
882, 48, 3, 8odinf 18178 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
8988iffalsed 4239 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = 0)
9087, 89eqtr4d 2795 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
91903expa 1112 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
922, 48odcl 18153 . . . 4 (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
9392adantl 473 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ0)
94 elnn0 11484 . . 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 1630  wcel 2137  wral 3048  ∃!wreu 3050  Vcvv 3338  wss 3713  ifcif 4228   class class class wbr 4802  cmpt 4879  ran crn 5265  wf 6043  1-1-ontowf1o 6046  cfv 6047  (class class class)co 6811  cen 8116  Fincfn 8119  cr 10125  0cc0 10126  1c1 10127   < clt 10264  cle 10265  cmin 10456  cn 11210  0cn0 11482  cz 11567  +crp 12023  ...cfz 12517   mod cmo 12860  chash 13309  cdvds 15180  Basecbs 16057  0gc0g 16300  Grpcgrp 17621  .gcmg 17739  odcod 18142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-omul 7732  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-sup 8511  df-inf 8512  df-oi 8578  df-card 8953  df-acn 8956  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-n0 11483  df-z 11568  df-uz 11878  df-rp 12024  df-fz 12518  df-fl 12785  df-mod 12861  df-seq 12994  df-exp 13053  df-hash 13310  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-dvds 15181  df-0g 16302  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-grp 17624  df-minusg 17625  df-sbg 17626  df-mulg 17740  df-od 18146
This theorem is referenced by:  oddvds2  18181  cyggenod  18484  cyggenod2  18485
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