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Theorem dfod2 19829

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 14156	. . . . 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 19351	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
5	4	3expa 1136	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
6	5	an32s 665	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
7	6	adantlr 728	. . . . . . . . 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 7146	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10		frn 6747	. . . . . . . 8 ⊢ (𝐹:ℤ⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	2	fvexi 6930	. . . . . . . . 9 ⊢ 𝑋 ∈ V
12	11	ssex 5289	. . . . . . . 8 ⊢ (ran 𝐹 ⊆ 𝑋 → ran 𝐹 ∈ V)
13	9, 10, 12	3syl 19	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ran 𝐹 ∈ V)
14		elfzelz 13696	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) → 𝑦 ∈ ℤ)
15	14	adantl 487	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
16		ovex 7484	. . . . . . . . . 10 ⊢ (𝑦 · 𝐴) ∈ V
17		oveq1 7458	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
18	8, 17	elrnmpt1s 5953	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ (𝑦 · 𝐴) ∈ V) → (𝑦 · 𝐴) ∈ ran 𝐹)
19	15, 16, 18	sylancl 598	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 · 𝐴) ∈ ran 𝐹)
20	19	ralrimiva 3158	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
21		zmodfz 14073	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
22	21	ancoms 464	. . . . . . . . . . . 12 ⊢ (((𝑂‘𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
23	22	adantll 727	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
24		simpllr 788	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℕ)
25		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑥 ∈ ℤ)
26	14	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
27		moddvds 16477	. . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . 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 12843	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℝ)
30	24	nnrpd 13202	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℝ⁺)
31		0z 12744	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℤ
32		nnz 12754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℤ)
33	32	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℤ)
34	33	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑂‘𝐴) ∈ ℤ)
35		elfzm11 13769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))))
36	31, 34, 35	sylancr 599	. . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 0 ≤ 𝑦)
39	37	simp3d 1162	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 < (𝑂‘𝐴))
40		modid 14076	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ (𝑂‘𝐴) ∈ ℝ⁺) ∧ (0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
41	29, 30, 38, 39, 40	syl22anc 852	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
42	41	eqeq2d 2774	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 14 ⊢ ((𝑥 mod (𝑂‘𝐴)) = 𝑦 ↔ 𝑦 = (𝑥 mod (𝑂‘𝐴)))
44	42, 43	bitrdi 290	. . . . . . . . . . . . 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 795	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝐺 ∈ Grp)
46		simp-4r 796	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
49	2, 47, 3, 48	odcong 19814	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑥 − 𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
50	45, 46, 25, 26, 49	syl112anc 1402	. . . . . . . . . . . . 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 313	. . . . . . . . . . . 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 3158	. . . . . . . . . . 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 3689	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
55	54	ralrimiva 3158	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
56		ovex 7484	. . . . . . . . . . 11 ⊢ (𝑥 · 𝐴) ∈ V
57	56	rgenw 3084	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
58		eqeq1 2767	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
59	58	reubidv 3385	. . . . . . . . . . 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 7126	. . . . . . . . . 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 237	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴))
63		eqid 2763	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴))
64	63	f1ompt 7143	. . . . . . . 8 ⊢ ((𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂‘𝐴) − 1))⟶1-1ontoran 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 595	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂‘𝐴) − 1))⟶1-1ontoran 𝐹)
66		f1oen2g 9033	. . . . . . 7 ⊢ (((0...((𝑂‘𝐴) − 1)) ∈ Fin ∧ ran 𝐹 ∈ V ∧ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)):(0...((𝑂‘𝐴) − 1))⟶1-1ontoran 𝐹) → (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹)
67	1, 13, 65, 66	syl3anc 1399	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹)
68		enfi 9240	. . . . . 6 ⊢ ((0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹 → ((0...((𝑂‘𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
69	67, 68	syl 18	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ((0...((𝑂‘𝐴) − 1)) ∈ Fin ↔ ran 𝐹 ∈ Fin))
70	1, 69	mpbid 235	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ran 𝐹 ∈ Fin)
71	70	iftrued 4493	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = (♯‘ran 𝐹))
72		fz01en 13726	. . . . . 6 ⊢ ((𝑂‘𝐴) ∈ ℤ → (0...((𝑂‘𝐴) − 1)) ≈ (1...(𝑂‘𝐴)))
73		ensym 9068	. . . . . 6 ⊢ ((0...((𝑂‘𝐴) − 1)) ≈ (1...(𝑂‘𝐴)) → (1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)))
74	33, 72, 73	3syl 19	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)))
75		entr 9071	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)) ∧ (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
76	74, 67, 75	syl2anc 596	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
77		fzfid 14156	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ∈ Fin)
78		hashen 14532	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂‘𝐴)) ≈ ran 𝐹))
79	77, 70, 78	syl2anc 596	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ((♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂‘𝐴)) ≈ ran 𝐹))
80	76, 79	mpbird 260	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹))
81		nnnn0 12653	. . . . 5 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℕ₀)
82	81	adantl 487	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℕ₀)
83		hashfz1 14531	. . . 4 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ → (♯‘(1...(𝑂‘𝐴))) = (𝑂‘𝐴))
84	82, 83	syl 18	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(1...(𝑂‘𝐴))) = (𝑂‘𝐴))
85	71, 80, 84	3eqtr2rd 2805	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
86		simp3 1156	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = 0)
87	2, 47, 3, 8	odinf 19828	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
88	87	iffalsed 4496	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = 0)
89	86, 88	eqtr4d 2801	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
90	89	3expa 1136	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
91	2, 47	odcl 19801	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
92	91	adantl 487	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
93		elnn0 12648	. . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
94	92, 93	sylib 221	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
95	85, 90, 94	mpjaodan 973	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃!wreu 3367 Vcvv 3454 ⊆ wss 3900 ifcif 4485 class class class wbr 5108 ↦ cmpt 5192 ran crn 5659 ⟶wf 6562 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 ≈ cen 9008 Fincfn 9011 ℝcr 11238 0cc0 11239 1c1 11240 < clt 11382 ≤ cle 11383 − cmin 11580 ℕcn 12375 ℕ₀cn0 12646 ℤcz 12733 ℝ⁺crp 13160 ...cfz 13679 mod cmo 14049 ♯chash 14515 ∥ cdvds 16466 Basecbs 17431 0_gc0g 17660 Grpcgrp 19194 ._gcmg 19327 odcod 19789

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-oadd 8518 df-omul 8519 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-acn 10060 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-rp 13161 df-fz 13680 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-0g 17662 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-od 19793

This theorem is used by: oddvds2 19831 cyggenod 20149 cyggenod2 20150 cycsubggenodd 20376 unitscyglem1 43511

W3C validator