Theorem idomsubgmo 44195

Description: The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Revised by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
idomsubgmo.g	⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))

Assertion

Ref	Expression
idomsubgmo	⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∃*𝑦 ∈ (SubGrp‘𝐺)(♯‘𝑦) = 𝑁)

Distinct variable groups:   𝑦,𝐺   𝑦,𝑁   𝑦,𝑅

Proof of Theorem idomsubgmo

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6898	. . . . . . . . 9 ⊢ (Base‘𝐺) ∈ V
2	1	rabex 5299	. . . . . . . 8 ⊢ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V
3		simp2l 1218	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ∈ (SubGrp‘𝐺))
4		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19337	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (SubGrp‘𝐺) → 𝑦 ⊆ (Base‘𝐺))
6	3, 5	syl 18	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ⊆ (Base‘𝐺))
7		simpl2l 1245	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → 𝑦 ∈ (SubGrp‘𝐺))
8		simp3l 1220	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = 𝑁)
9		simp1r 1217	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ)
10	9	nnnn0d 12667	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ0)
11	8, 10	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) ∈ ℕ0)
12		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
13		hashclb 14502	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ V → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ0))
14	12, 13	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ0)
15	11, 14	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ∈ Fin)
16	15	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → 𝑦 ∈ Fin)
17		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑦)
18		eqid 2760	. . . . . . . . . . . . 13 ⊢ (od‘𝐺) = (od‘𝐺)
19	18	odsubdvds 19785	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ 𝑦) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑦))
20	7, 16, 17, 19	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑦))
21	8	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → (♯‘𝑦) = 𝑁)
22	20, 21	breqtrd 5130	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → ((od‘𝐺)‘𝑧) ∥ 𝑁)
23	6, 22	ssrabdv 3998	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ⊆ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁})
24		simp2r 1219	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ∈ (SubGrp‘𝐺))
25	4	subgss 19337	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
26	24, 25	syl 18	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ⊆ (Base‘𝐺))
27		simpl2r 1246	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (SubGrp‘𝐺))
28		simp3r 1221	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) = 𝑁)
29	28, 10	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) ∈ ℕ0)
30		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
31		hashclb 14502	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin ↔ (♯‘𝑥) ∈ ℕ0))
32	30, 31	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Fin ↔ (♯‘𝑥) ∈ ℕ0)
33	29, 32	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ∈ Fin)
34	33	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ Fin)
35		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
36	18	odsubdvds 19785	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ Fin ∧ 𝑧 ∈ 𝑥) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑥))
37	27, 34, 35, 36	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑥))
38	28	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → (♯‘𝑥) = 𝑁)
39	37, 38	breqtrd 5130	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → ((od‘𝐺)‘𝑧) ∥ 𝑁)
40	26, 39	ssrabdv 3998	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ⊆ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁})
41	23, 40	unssd 4190	. . . . . . . 8 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ⊆ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁})
42		ssdomg 9027	. . . . . . . 8 ⊢ ({𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V → ((𝑦 ∪ 𝑥) ⊆ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} → (𝑦 ∪ 𝑥) ≼ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}))
43	2, 41, 42	mpsyl 69	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ≼ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁})
44		idomsubgmo.g	. . . . . . . . . . 11 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))
45	44, 4, 18	idomodle 44193	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → (♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ 𝑁)
46	45	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ 𝑁)
47	46, 8	breqtrrd 5132	. . . . . . . 8 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ (♯‘𝑦))
48	2	a1i 11	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V)
49		hashbnd 14480	. . . . . . . . . 10 ⊢ (({𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V ∧ (♯‘𝑦) ∈ ℕ0 ∧ (♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ (♯‘𝑦)) → {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ Fin)
50	48, 11, 47, 49	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ Fin)
51		hashdom 14523	. . . . . . . . 9 ⊢ (({𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ Fin ∧ 𝑦 ∈ V) → ((♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ (♯‘𝑦) ↔ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ≼ 𝑦))
52	50, 12, 51	sylancl 598	. . . . . . . 8 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → ((♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ (♯‘𝑦) ↔ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ≼ 𝑦))
53	47, 52	mpbid 235	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ≼ 𝑦)
54		domtr 9034	. . . . . . 7 ⊢ (((𝑦 ∪ 𝑥) ≼ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∧ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ≼ 𝑦) → (𝑦 ∪ 𝑥) ≼ 𝑦)
55	43, 53, 54	syl2anc 596	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ≼ 𝑦)
56	12, 30	unex 7761	. . . . . . 7 ⊢ (𝑦 ∪ 𝑥) ∈ V
57		ssun1 4176	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝑥)
58		ssdomg 9027	. . . . . . 7 ⊢ ((𝑦 ∪ 𝑥) ∈ V → (𝑦 ⊆ (𝑦 ∪ 𝑥) → 𝑦 ≼ (𝑦 ∪ 𝑥)))
59	56, 57, 58	mp2 9	. . . . . 6 ⊢ 𝑦 ≼ (𝑦 ∪ 𝑥)
60		sbth 9116	. . . . . 6 ⊢ (((𝑦 ∪ 𝑥) ≼ 𝑦 ∧ 𝑦 ≼ (𝑦 ∪ 𝑥)) → (𝑦 ∪ 𝑥) ≈ 𝑦)
61	55, 59, 60	sylancl 598	. . . . 5 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ≈ 𝑦)
62	8, 28	eqtr4d 2798	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = (♯‘𝑥))
63		hashen 14491	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ 𝑥 ∈ Fin) → ((♯‘𝑦) = (♯‘𝑥) ↔ 𝑦 ≈ 𝑥))
64	15, 33, 63	syl2anc 596	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → ((♯‘𝑦) = (♯‘𝑥) ↔ 𝑦 ≈ 𝑥))
65	62, 64	mpbid 235	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ≈ 𝑥)
66		fiuneneq 44194	. . . . . 6 ⊢ ((𝑦 ≈ 𝑥 ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ 𝑥) ≈ 𝑦 ↔ 𝑦 = 𝑥))
67	65, 15, 66	syl2anc 596	. . . . 5 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → ((𝑦 ∪ 𝑥) ≈ 𝑦 ↔ 𝑦 = 𝑥))
68	61, 67	mpbid 235	. . . 4 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 = 𝑥)
69	68	3expia 1139	. . 3 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺))) → (((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
70	69	ralrimivva 3205	. 2 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
71		fveqeq2 6894	. . 3 ⊢ (𝑦 = 𝑥 → ((♯‘𝑦) = 𝑁 ↔ (♯‘𝑥) = 𝑁))
72	71	rmo4 3687	. 2 ⊢ (∃*𝑦 ∈ (SubGrp‘𝐺)(♯‘𝑦) = 𝑁 ↔ ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
73	70, 72	sylibr 237	1 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∃*𝑦 ∈ (SubGrp‘𝐺)(♯‘𝑦) = 𝑁)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃*wrmo 3364  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420   ≈ cen 8970   ≼ cdom 8971  Fincfn 8973   ≤ cle 11344  ℕcn 12335  ℕ₀cn0 12606  ♯chash 14474   ∥ cdvds 16422  Basecbs 17387   ↾_s cress 17408  SubGrpcsubg 19330  odcod 19738  mulGrpcmgp 20360  Unitcui 20585  IDomncidom 20945

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278  ax-addf 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-omul 8481  df-er 8717  df-ec 8719  df-qs 8723  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-sup 9434  df-inf 9435  df-oi 9504  df-dju 9982  df-card 10020  df-acn 10023  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-xnn0 12680  df-z 12694  df-dec 12815  df-uz 12966  df-rp 13121  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-sum 15854  df-dvds 16423  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-eqg 19335  df-ghm 19428  df-cntz 19531  df-od 19742  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-srg 20413  df-ring 20461  df-cring 20462  df-oppr 20567  df-dvdsr 20587  df-unit 20588  df-invr 20618  df-rhm 20702  df-nzr 20763  df-subrng 20798  df-subrg 20822  df-rlreg 20946  df-domn 20947  df-idom 20948  df-lmod 21137  df-lss 21207  df-lsp 21247  df-cnfld 21679  df-assa 22161  df-asp 22162  df-ascl 22163  df-psr 22217  df-mvr 22218  df-mpl 22219  df-opsr 22221  df-evls 22383  df-evl 22384  df-psr1 22498  df-vr1 22499  df-ply1 22500  df-coe1 22501  df-evl1 22634  df-mdeg 26373  df-deg1 26374  df-mon1 26449  df-uc1p 26450  df-q1p 26451  df-r1p 26452

This theorem is used by:  proot1mul  44196