dfod2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dfod2		Structured version Visualization version GIF version

Theorem dfod2 18691

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 13342	. . . . 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 18245	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
5	4	3expa 1114	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
6	5	an32s 650	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
7	6	adantlr 713	. . . . . . . . 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 6878	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → 𝐹:ℤ⟶𝑋)
10		frn 6520	. . . . . . . 8 ⊢ (𝐹:ℤ⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	2	fvexi 6684	. . . . . . . . 9 ⊢ 𝑋 ∈ V
12	11	ssex 5225	. . . . . . . 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 12909	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) → 𝑦 ∈ ℤ)
15	14	adantl 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
16		ovex 7189	. . . . . . . . . 10 ⊢ (𝑦 · 𝐴) ∈ V
17		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐴) = (𝑦 · 𝐴))
18	8, 17	elrnmpt1s 5829	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ (𝑦 · 𝐴) ∈ V) → (𝑦 · 𝐴) ∈ ran 𝐹)
19	15, 16, 18	sylancl 588	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 · 𝐴) ∈ ran 𝐹)
20	19	ralrimiva 3182	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑦 · 𝐴) ∈ ran 𝐹)
21		zmodfz 13262	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
22	21	ancoms 461	. . . . . . . . . . . 12 ⊢ (((𝑂‘𝐴) ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
23	22	adantll 712	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑥 mod (𝑂‘𝐴)) ∈ (0...((𝑂‘𝐴) − 1)))
24		simpllr 774	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℕ)
25		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑥 ∈ ℤ)
26	14	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℤ)
27		moddvds 15618	. . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . 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 12088	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 ∈ ℝ)
30	24	nnrpd 12430	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑂‘𝐴) ∈ ℝ⁺)
31		0z 11993	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℤ
32		nnz 12005	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℤ)
33	32	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℤ)
34	33	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → (𝑂‘𝐴) ∈ ℤ)
35		elfzm11 12979	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))))
36	31, 34, 35	sylancr 589	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 0 ≤ 𝑦)
39	37	simp3d 1140	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝑦 < (𝑂‘𝐴))
40		modid 13265	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℝ ∧ (𝑂‘𝐴) ∈ ℝ⁺) ∧ (0 ≤ 𝑦 ∧ 𝑦 < (𝑂‘𝐴))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
41	29, 30, 38, 39, 40	syl22anc 836	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → (𝑦 mod (𝑂‘𝐴)) = 𝑦)
42	41	eqeq2d 2832	. . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . 14 ⊢ ((𝑥 mod (𝑂‘𝐴)) = 𝑦 ↔ 𝑦 = (𝑥 mod (𝑂‘𝐴)))
44	42, 43	syl6bb 289	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ (0...((𝑂‘𝐴) − 1))) → 𝐺 ∈ Grp)
46		simp-4r 782	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
49	2, 47, 3, 48	odcong 18677	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑥 − 𝑦) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
50	45, 46, 25, 26, 49	syl112anc 1370	. . . . . . . . . . . . 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 312	. . . . . . . . . . . 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 3182	. . . . . . . . . . 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 3719	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
55	54	ralrimiva 3182	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑥 ∈ ℤ ∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))(𝑥 · 𝐴) = (𝑦 · 𝐴))
56		ovex 7189	. . . . . . . . . . 11 ⊢ (𝑥 · 𝐴) ∈ V
57	56	rgenw 3150	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℤ (𝑥 · 𝐴) ∈ V
58		eqeq1 2825	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥 · 𝐴) → (𝑧 = (𝑦 · 𝐴) ↔ (𝑥 · 𝐴) = (𝑦 · 𝐴)))
59	58	reubidv 3389	. . . . . . . . . . 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 6860	. . . . . . . . . 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 236	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ∀𝑧 ∈ ran 𝐹∃!𝑦 ∈ (0...((𝑂‘𝐴) − 1))𝑧 = (𝑦 · 𝐴))
63		eqid 2821	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴)) = (𝑦 ∈ (0...((𝑂‘𝐴) − 1)) ↦ (𝑦 · 𝐴))
64	63	f1ompt 6875	. . . . . . . 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 585	. . . . . . 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 8526	. . . . . . 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 1367	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹)
68		enfi 8734	. . . . . 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 234	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ran 𝐹 ∈ Fin)
71	70	iftrued 4475	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = (♯‘ran 𝐹))
72		fz01en 12936	. . . . . 6 ⊢ ((𝑂‘𝐴) ∈ ℤ → (0...((𝑂‘𝐴) − 1)) ≈ (1...(𝑂‘𝐴)))
73		ensym 8558	. . . . . 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 8561	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ≈ (0...((𝑂‘𝐴) − 1)) ∧ (0...((𝑂‘𝐴) − 1)) ≈ ran 𝐹) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
76	74, 67, 75	syl2anc 586	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ≈ ran 𝐹)
77		fzfid 13342	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (1...(𝑂‘𝐴)) ∈ Fin)
78		hashen 13708	. . . . 5 ⊢ (((1...(𝑂‘𝐴)) ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂‘𝐴)) ≈ ran 𝐹))
79	77, 70, 78	syl2anc 586	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → ((♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹) ↔ (1...(𝑂‘𝐴)) ≈ ran 𝐹))
80	76, 79	mpbird 259	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(1...(𝑂‘𝐴))) = (♯‘ran 𝐹))
81		nnnn0 11905	. . . . 5 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℕ₀)
82	81	adantl 484	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) ∈ ℕ₀)
83		hashfz1 13707	. . . 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 2863	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
86		simp3 1134	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = 0)
87	2, 47, 3, 8	odinf 18690	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
88	87	iffalsed 4478	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0) = 0)
89	86, 88	eqtr4d 2859	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
90	89	3expa 1114	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑂‘𝐴) = 0) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
91	2, 47	odcl 18664	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
92	91	adantl 484	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ₀)
93		elnn0 11900	. . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
94	92, 93	sylib 220	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
95	85, 90, 94	mpjaodan 955	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃!wreu 3140 Vcvv 3494 ⊆ wss 3936 ifcif 4467 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ≈ cen 8506 Fincfn 8509 ℝcr 10536 0cc0 10537 1c1 10538 < clt 10675 ≤ cle 10676 − cmin 10870 ℕcn 11638 ℕ₀cn0 11898 ℤcz 11982 ℝ⁺crp 12390 ...cfz 12893 mod cmo 13238 ♯chash 13691 ∥ cdvds 15607 Basecbs 16483 0_gc0g 16713 Grpcgrp 18103 ._gcmg 18224 odcod 18652

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-od 18656

This theorem is referenced by: oddvds2 18693 cyggenod 19003 cyggenod2 19004 cycsubggenodd 19231

W3C validator