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Theorem dfod2 19582

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	⊢ · = (._g‘𝐺)
odf1.4	⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))

**Assertion**
Ref	Expression
dfod2	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 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.

Step	Hyp	Ref	Expression
1		fzfid 14014	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (0...((𝑂‘𝐴) − 1)) ∈ Fin)
2		odf1.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = (Base‘𝐺)
3		odf1.3	. . . . . . . . . . . . 13 ⊢ · = (._g‘𝐺)
4	2, 3	mulgcl 19109	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
5	4	3expa 1119	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
6	5	an32s 652	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
7	6	adantlr 715	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
8		odf1.4	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
9	7, 8	fmptd 7134	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10		frn 6743	. . . . . . . 8 ⊢ (𝐹:ℤ⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	2	fvexi 6920	. . . . . . . . 9 ⊢ 𝑋 ∈ V
12	11	ssex 5321	. . . . . . . 8 ⊢ (ran 𝐹 ⊆ 𝑋 → ran 𝐹 ∈ V)
13	9, 10, 12	3syl 18	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ran 𝐹 ∈ V)
14		elfzelz 13564	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) → 𝑦 ∈ ℤ)
15	14	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
16		ovex 7464	. . . . . . . . . 10 ⊢ (𝑦 · 𝐴) ∈ V
17		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
18	8, 17	elrnmpt1s 5970	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ (𝑦 · 𝐴) ∈ V) → (𝑦 · 𝐴) ∈ ran 𝐹)
19	15, 16, 18	sylancl 586	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 · 𝐴) ∈ ran 𝐹)
20	19	ralrimiva 3146	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
21		zmodfz 13933	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
22	21	ancoms 458	. . . . . . . . . . . 12 ⊢ (((𝑂‘𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
23	22	adantll 714	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
24		simpllr 776	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℕ)
25		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑥 ∈ ℤ)
26	14	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
27		moddvds 16301	. . . . . . . . . . . . . 14 ⊢ (((𝑂‘𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 mod (𝑂‘𝐴)) = (𝑦 mod (𝑂‘𝐴)) ↔ (𝑂‘𝐴) ∥ (𝑥 − 𝑦)))
28	24, 25, 26, 27	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → ((𝑥 mod (𝑂‘𝐴)) = (𝑦 mod (𝑂‘𝐴)) ↔ (𝑂‘𝐴) ∥ (𝑥 − 𝑦)))
29	26	zred 12722	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℝ)
30	24	nnrpd 13075	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℝ⁺)
31		0z 12624	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℤ
32		nnz 12634	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℤ)
33	32	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℤ)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑂‘𝐴) ∈ ℤ)
35		elfzm11 13635	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))))
36	31, 34, 35	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))))
37	36	biimpa 476	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴)))
38	37	simp2d 1144	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 0 ≤ 𝑦)
39	37	simp3d 1145	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 < (𝑂‘𝐴))
40		modid 13936	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ (𝑂‘𝐴) ∈ ℝ⁺) ∧ (0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
41	29, 30, 38, 39, 40	syl22anc 839	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
42	41	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → ((𝑥 mod (𝑂‘𝐴)) = (𝑦 mod (𝑂‘𝐴)) ↔ (𝑥 mod (𝑂‘𝐴)) = 𝑦))
43		eqcom 2744	. . . . . . . . . . . . . 14 ⊢ ((𝑥 mod (𝑂‘𝐴)) = 𝑦 ↔ 𝑦 = (𝑥 mod (𝑂‘𝐴)))
44	42, 43	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → ((𝑥 mod (𝑂‘𝐴)) = (𝑦 mod (𝑂‘𝐴)) ↔ 𝑦 = (𝑥 mod (𝑂‘𝐴))))
45		simp-4l 783	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝐺 ∈ Grp)
46		simp-4r 784	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝐴 ∈ 𝑋)
47		odf1.2	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (od‘𝐺)
48		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
49	2, 47, 3, 48	odcong 19567	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑥 − 𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
50	45, 46, 25, 26, 49	syl112anc 1376	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → ((𝑂‘𝐴) ∥ (𝑥 − 𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
51	28, 44, 50	3bitr3rd 310	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → ((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂‘𝐴))))
52	51	ralrimiva 3146	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∀𝑦 ∈ (0...((𝑂‘𝐴) − 1))((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂‘𝐴))))
53		reu6i 3734	. . . . . . . . . . 11 ⊢ (((𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)) ∧ ∀𝑦 ∈ (0...((𝑂‘𝐴) − 1))((𝑥 · 𝐴) = (𝑦 · 𝐴) ↔ 𝑦 = (𝑥 mod (𝑂‘𝐴)))) → ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
54	23, 52, 53	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
55	54	ralrimiva 3146	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
56		ovex 7464	. . . . . . . . . . 11 ⊢ (𝑥 · 𝐴) ∈ V
57	56	rgenw 3065	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
58		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
59	58	reubidv 3398	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥 · 𝐴) → (∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴)))
60	8, 59	ralrnmptw 7114	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V → (∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴)))
61	57, 60	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴) ↔ ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
62	55, 61	sylibr 234	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴))
63		eqid 2737	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴))
64	63	f1ompt 7131	. . . . . . . 8 ⊢ ((𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂‘𝐴) − 1))–1-1-onto→ran 𝐹 ↔ (∀𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹 ∧ ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴)))
65	20, 62, 64	sylanbrc 583	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂‘𝐴) − 1))–1-1-onto→ran 𝐹)
66		f1oen2g 9009	. . . . . . 7 ⊢ (((0...((𝑂‘𝐴) − 1)) ∈ Fin ∧ ran 𝐹 ∈ V ∧ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂‘𝐴) − 1))–1-1-onto→ran 𝐹) → (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹)
67	1, 13, 65, 66	syl3anc 1373	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹)
68		enfi 9227	. . . . . 6 ⊢ ((0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹 → ((0...((𝑂‘𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
69	67, 68	syl 17	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ((0...((𝑂‘𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
70	1, 69	mpbid 232	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ran 𝐹 ∈ Fin)
71	70	iftrued 4533	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = (♯‘ran 𝐹))
72		fz01en 13592	. . . . . 6 ⊢ ((𝑂‘𝐴) ∈ ℤ → (0...((𝑂‘𝐴) − 1)) ≈ (1...(𝑂‘𝐴)))
73		ensym 9043	. . . . . 6 ⊢ ((0...((𝑂‘𝐴) − 1)) ≈ (1...(𝑂‘𝐴)) → (1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)))
74	33, 72, 73	3syl 18	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)))
75		entr 9046	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)) ∧ (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
76	74, 67, 75	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
77		fzfid 14014	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ∈ Fin)
78		hashen 14386	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂‘𝐴)) ≈ ran 𝐹))
79	77, 70, 78	syl2anc 584	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ((♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂‘𝐴)) ≈ ran 𝐹))
80	76, 79	mpbird 257	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹))
81		nnnn0 12533	. . . . 5 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℕ₀)
82	81	adantl 481	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℕ₀)
83		hashfz1 14385	. . . 4 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ → (♯‘(1...(𝑂‘𝐴))) = (𝑂‘𝐴))
84	82, 83	syl 17	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(1...(𝑂‘𝐴))) = (𝑂‘𝐴))
85	71, 80, 84	3eqtr2rd 2784	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
86		simp3 1139	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = 0)
87	2, 47, 3, 8	odinf 19581	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
88	87	iffalsed 4536	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = 0)
89	86, 88	eqtr4d 2780	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
90	89	3expa 1119	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
91	2, 47	odcl 19554	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
92	91	adantl 481	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
93		elnn0 12528	. . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
94	92, 93	sylib 218	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
95	85, 90, 94	mpjaodan 961	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 Vcvv 3480 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ≈ cen 8982 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ℝ⁺crp 13034 ...cfz 13547 mod cmo 13909 ♯chash 14369 ∥ cdvds 16290 Basecbs 17247 0_gc0g 17484 Grpcgrp 18951 ._gcmg 19085 odcod 19542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-od 19546

This theorem is referenced by: oddvds2 19584 cyggenod 19902 cyggenod2 19903 cycsubggenodd 20129 unitscyglem1 42196

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