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Theorem dfod2 17804
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 12634 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ∈ Fin)
2 odf1.1 . . . . . . . . . . . . 13 𝑋 = (Base‘𝐺)
3 odf1.3 . . . . . . . . . . . . 13 · = (.g𝐺)
42, 3mulgcl 17382 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
543expa 1257 . . . . . . . . . . 11 (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
65an32s 842 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
76adantlr 747 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
8 odf1.4 . . . . . . . . 9 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
97, 8fmptd 6292 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10 frn 5966 . . . . . . . 8 (𝐹:ℤ⟶𝑋 → ran 𝐹𝑋)
11 fvex 6113 . . . . . . . . . 10 (Base‘𝐺) ∈ V
122, 11eqeltri 2684 . . . . . . . . 9 𝑋 ∈ V
1312ssex 4730 . . . . . . . 8 (ran 𝐹𝑋 → ran 𝐹 ∈ V)
149, 10, 133syl 18 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ran 𝐹 ∈ V)
15 elfzelz 12213 . . . . . . . . . . 11 (𝑦 ∈ (0...((𝑂𝐴) − 1)) → 𝑦 ∈ ℤ)
1615adantl 481 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℤ)
17 ovex 6577 . . . . . . . . . 10 (𝑦 · 𝐴) ∈ V
18 oveq1 6556 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
198, 18elrnmpt1s 5294 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ (𝑦 · 𝐴) ∈ V) → (𝑦 · 𝐴) ∈ ran 𝐹)
2016, 17, 19sylancl 693 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 · 𝐴) ∈ ran 𝐹)
2120ralrimiva 2949 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
22 zmodfz 12554 . . . . . . . . . . . . 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 746 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂𝐴)) ∈ (0...((𝑂𝐴) − 1)))
25 simpllr 795 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑂𝐴) ∈ ℕ)
26 simplr 788 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑥 ∈ ℤ)
2715adantl 481 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℤ)
28 moddvds 14829 . . . . . . . . . . . . . 14 (((𝑂𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
2925, 26, 27, 28syl3anc 1318 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑂𝐴) ∥ (𝑥𝑦)))
3027zred 11358 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 ∈ ℝ)
3125nnrpd 11746 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑂𝐴) ∈ ℝ+)
32 0z 11265 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℤ
33 nnz 11276 . . . . . . . . . . . . . . . . . . . . 21 ((𝑂𝐴) ∈ ℕ → (𝑂𝐴) ∈ ℤ)
3433adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) ∈ ℤ)
3534adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑂𝐴) ∈ ℤ)
36 elfzm11 12280 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℤ ∧ (𝑂𝐴) ∈ ℤ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴))))
3732, 35, 36sylancr 694 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴))))
3837biimpa 500 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦𝑦 < (𝑂𝐴)))
3938simp2d 1067 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 0 ≤ 𝑦)
4038simp3d 1068 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝑦 < (𝑂𝐴))
41 modid 12557 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℝ ∧ (𝑂𝐴) ∈ ℝ+) ∧ (0 ≤ 𝑦𝑦 < (𝑂𝐴))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4230, 31, 39, 40, 41syl22anc 1319 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → (𝑦 mod (𝑂𝐴)) = 𝑦)
4342eqeq2d 2620 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 mod (𝑂𝐴)) = (𝑦 mod (𝑂𝐴)) ↔ (𝑥 mod (𝑂𝐴)) = 𝑦))
44 eqcom 2617 . . . . . . . . . . . . . 14 ((𝑥 mod (𝑂𝐴)) = 𝑦𝑦 = (𝑥 mod (𝑂𝐴)))
4543, 44syl6bb 275 . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝐺 ∈ Grp)
47 simp-4r 803 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → 𝐴𝑋)
48 odf1.2 . . . . . . . . . . . . . . 15 𝑂 = (od‘𝐺)
49 eqid 2610 . . . . . . . . . . . . . . 15 (0g𝐺) = (0g𝐺)
502, 48, 3, 49odcong 17791 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂𝐴) ∥ (𝑥𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5146, 47, 26, 27, 50syl112anc 1322 . . . . . . . . . . . . 13 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑂𝐴) ∥ (𝑥𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
5229, 45, 513bitr3rd 298 . . . . . . . . . . . 12 (((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂𝐴) − 1))) → ((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
5352ralrimiva 2949 . . . . . . . . . . 11 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∀𝑦 ∈ (0...((𝑂𝐴) − 1))((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂𝐴))))
54 reu6i 3364 . . . . . . . . . . 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 691 . . . . . . . . . 10 ((((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
5655ralrimiva 2949 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
57 ovex 6577 . . . . . . . . . . 11 (𝑥 · 𝐴) ∈ V
5857rgenw 2908 . . . . . . . . . 10 𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
59 eqeq1 2614 . . . . . . . . . . . 12 (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
6059reubidv 3103 . . . . . . . . . . 11 (𝑧 = (𝑥 · 𝐴) → (∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∃!𝑦 ∈ (0...((𝑂𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴)))
618, 60ralrnmpt 6276 . . . . . . . . . 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 223 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂𝐴) − 1))𝑧 = (𝑦 · 𝐴))
64 eqid 2610 . . . . . . . . 9 (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴))
6564f1ompt 6290 . . . . . . . 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 695 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑦 ∈ (0...((𝑂𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂𝐴) − 1))–1-1-onto→ran 𝐹)
67 f1oen2g 7858 . . . . . . 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 1318 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (0...((𝑂𝐴) − 1)) ≈ ran 𝐹)
69 enfi 8061 . . . . . 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 221 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ran 𝐹 ∈ Fin)
7271iftrued 4044 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0) = (#‘ran 𝐹))
73 fz01en 12240 . . . . . 6 ((𝑂𝐴) ∈ ℤ → (0...((𝑂𝐴) − 1)) ≈ (1...(𝑂𝐴)))
74 ensym 7891 . . . . . 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 7894 . . . . 5 (((1...(𝑂𝐴)) ≈ (0...((𝑂𝐴) − 1)) ∧ (0...((𝑂𝐴) − 1)) ≈ ran 𝐹) → (1...(𝑂𝐴)) ≈ ran 𝐹)
7775, 68, 76syl2anc 691 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ≈ ran 𝐹)
78 fzfid 12634 . . . . 5 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (1...(𝑂𝐴)) ∈ Fin)
79 hashen 12997 . . . . 5 (((1...(𝑂𝐴)) ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((#‘(1...(𝑂𝐴))) = (#‘ran 𝐹) ↔ (1...(𝑂𝐴)) ≈ ran 𝐹))
8078, 71, 79syl2anc 691 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → ((#‘(1...(𝑂𝐴))) = (#‘ran 𝐹) ↔ (1...(𝑂𝐴)) ≈ ran 𝐹))
8177, 80mpbird 246 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (#‘(1...(𝑂𝐴))) = (#‘ran 𝐹))
82 nnnn0 11176 . . . . 5 ((𝑂𝐴) ∈ ℕ → (𝑂𝐴) ∈ ℕ0)
8382adantl 481 . . . 4 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) ∈ ℕ0)
84 hashfz1 12996 . . . 4 ((𝑂𝐴) ∈ ℕ0 → (#‘(1...(𝑂𝐴))) = (𝑂𝐴))
8583, 84syl 17 . . 3 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (#‘(1...(𝑂𝐴))) = (𝑂𝐴))
8672, 81, 853eqtr2rd 2651 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) ∈ ℕ) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0))
87 simp3 1056 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = 0)
882, 48, 3, 8odinf 17803 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
8988iffalsed 4047 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0) = 0)
9087, 89eqtr4d 2647 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0))
91903expa 1257 . 2 (((𝐺 ∈ Grp ∧ 𝐴𝑋) ∧ (𝑂𝐴) = 0) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0))
922, 48odcl 17778 . . . 4 (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
9392adantl 481 . . 3 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ0)
94 elnn0 11171 . . 3 ((𝑂𝐴) ∈ ℕ0 ↔ ((𝑂𝐴) ∈ ℕ ∨ (𝑂𝐴) = 0))
9593, 94sylib 207 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) ∈ ℕ ∨ (𝑂𝐴) = 0))
9686, 91, 95mpjaodan 823 1 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  ∃!wreu 2898  Vcvv 3173  wss 3540  ifcif 4036   class class class wbr 4583  cmpt 4643  ran crn 5039  wf 5800  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  cen 7838  Fincfn 7841  cr 9814  0cc0 9815  1c1 9816   < clt 9953  cle 9954  cmin 10145  cn 10897  0cn0 11169  cz 11254  +crp 11708  ...cfz 12197   mod cmo 12530  #chash 12979  cdvds 14821  Basecbs 15695  0gc0g 15923  Grpcgrp 17245  .gcmg 17363  odcod 17767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-od 17771
This theorem is referenced by:  oddvds2  17806  cyggenod  18109  cyggenod2  18110
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