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Theorem dfod2 19432

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 13938	. . . . 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 18971	. . . . . . . . . . . 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 651	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
7	6	adantlr 714	. . . . . . . . 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 7114	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10		frn 6725	. . . . . . . 8 ⊢ (𝐹:ℤ⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	2	fvexi 6906	. . . . . . . . 9 ⊢ 𝑋 ∈ V
12	11	ssex 5322	. . . . . . . 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 13501	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) → 𝑦 ∈ ℤ)
15	14	adantl 483	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
16		ovex 7442	. . . . . . . . . 10 ⊢ (𝑦 · 𝐴) ∈ V
17		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
18	8, 17	elrnmpt1s 5957	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ (𝑦 · 𝐴) ∈ V) → (𝑦 · 𝐴) ∈ ran 𝐹)
19	15, 16, 18	sylancl 587	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 · 𝐴) ∈ ran 𝐹)
20	19	ralrimiva 3147	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
21		zmodfz 13858	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
22	21	ancoms 460	. . . . . . . . . . . 12 ⊢ (((𝑂‘𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
23	22	adantll 713	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
24		simpllr 775	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℕ)
25		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑥 ∈ ℤ)
26	14	adantl 483	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
27		moddvds 16208	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . 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 12666	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℝ)
30	24	nnrpd 13014	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℝ⁺)
31		0z 12569	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℤ
32		nnz 12579	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℤ)
33	32	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℤ)
34	33	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑂‘𝐴) ∈ ℤ)
35		elfzm11 13572	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))))
36	31, 34, 35	sylancr 588	. . . . . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . . . . . 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 13861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ (𝑂‘𝐴) ∈ ℝ⁺) ∧ (0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
41	29, 30, 38, 39, 40	syl22anc 838	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
42	41	eqeq2d 2744	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝐺 ∈ Grp)
46		simp-4r 783	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
49	2, 47, 3, 48	odcong 19417	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑥 − 𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
50	45, 46, 25, 26, 49	syl112anc 1375	. . . . . . . . . . . . 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 3147	. . . . . . . . . . 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 3725	. . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
55	54	ralrimiva 3147	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
56		ovex 7442	. . . . . . . . . . 11 ⊢ (𝑥 · 𝐴) ∈ V
57	56	rgenw 3066	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
58		eqeq1 2737	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
59	58	reubidv 3395	. . . . . . . . . . 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 7096	. . . . . . . . . 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 233	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴))
63		eqid 2733	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴))
64	63	f1ompt 7111	. . . . . . . 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 584	. . . . . . 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 8964	. . . . . . 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 1372	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹)
68		enfi 9190	. . . . . 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 231	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ran 𝐹 ∈ Fin)
71	70	iftrued 4537	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = (♯‘ran 𝐹))
72		fz01en 13529	. . . . . 6 ⊢ ((𝑂‘𝐴) ∈ ℤ → (0...((𝑂‘𝐴) − 1)) ≈ (1...(𝑂‘𝐴)))
73		ensym 8999	. . . . . 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 9002	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)) ∧ (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
76	74, 67, 75	syl2anc 585	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
77		fzfid 13938	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ∈ Fin)
78		hashen 14307	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂‘𝐴)) ≈ ran 𝐹))
79	77, 70, 78	syl2anc 585	. . . 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 12479	. . . . 5 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℕ₀)
82	81	adantl 483	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℕ₀)
83		hashfz1 14306	. . . 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 2780	. 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 19431	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
88	87	iffalsed 4540	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = 0)
89	86, 88	eqtr4d 2776	. . 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 19404	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
92	91	adantl 483	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
93		elnn0 12474	. . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
94	92, 93	sylib 217	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
95	85, 90, 94	mpjaodan 958	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 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃!wreu 3375 Vcvv 3475 ⊆ wss 3949 ifcif 4529 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 ≈ cen 8936 Fincfn 8939 ℝcr 11109 0cc0 11110 1c1 11111 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ℝ⁺crp 12974 ...cfz 13484 mod cmo 13834 ♯chash 14290 ∥ cdvds 16197 Basecbs 17144 0_gc0g 17385 Grpcgrp 18819 ._gcmg 18950 odcod 19392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-od 19396

This theorem is referenced by: oddvds2 19434 cyggenod 19752 cyggenod2 19753 cycsubggenodd 19979

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