Theorem idomsubgmo 43472

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 6846	. . . . . . . . 9 ⊢ (Base‘𝐺) ∈ V
2	1	rabex 5283	. . . . . . . 8 ⊢ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V
3		simp2l 1201	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ∈ (SubGrp‘𝐺))
4		eqid 2735	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19059	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (SubGrp‘𝐺) → 𝑦 ⊆ (Base‘𝐺))
6	3, 5	syl 17	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ⊆ (Base‘𝐺))
7		simpl2l 1228	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → 𝑦 ∈ (SubGrp‘𝐺))
8		simp3l 1203	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = 𝑁)
9		simp1r 1200	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ)
10	9	nnnn0d 12464	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ0)
11	8, 10	eqeltrd 2835	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) ∈ ℕ0)
12		vex 3443	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
13		hashclb 14283	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ V → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ0))
14	12, 13	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ0)
15	11, 14	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ∈ Fin)
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → 𝑦 ∈ Fin)
17		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑦)
18		eqid 2735	. . . . . . . . . . . . 13 ⊢ (od‘𝐺) = (od‘𝐺)
19	18	odsubdvds 19502	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ 𝑦) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑦))
20	7, 16, 17, 19	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑦))
21	8	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → (♯‘𝑦) = 𝑁)
22	20, 21	breqtrd 5123	. . . . . . . . . 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 4024	. . . . . . . . 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 1202	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ∈ (SubGrp‘𝐺))
25	4	subgss 19059	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ⊆ (Base‘𝐺))
27		simpl2r 1229	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (SubGrp‘𝐺))
28		simp3r 1204	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) = 𝑁)
29	28, 10	eqeltrd 2835	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) ∈ ℕ0)
30		vex 3443	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
31		hashclb 14283	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin ↔ (♯‘𝑥) ∈ ℕ0))
32	30, 31	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Fin ↔ (♯‘𝑥) ∈ ℕ0)
33	29, 32	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ∈ Fin)
34	33	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ Fin)
35		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
36	18	odsubdvds 19502	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ Fin ∧ 𝑧 ∈ 𝑥) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑥))
37	27, 34, 35, 36	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑥))
38	28	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → (♯‘𝑥) = 𝑁)
39	37, 38	breqtrd 5123	. . . . . . . . . 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 4024	. . . . . . . . 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 4143	. . . . . . . 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 8939	. . . . . . . 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 68	. . . . . . 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 43470	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → (♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ 𝑁)
46	45	3ad2ant1 1134	. . . . . . . . 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 5125	. . . . . . . 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 14261	. . . . . . . . . 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 1374	. . . . . . . . 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 14304	. . . . . . . . 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 587	. . . . . . . 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 232	. . . . . . 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 8946	. . . . . . 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 585	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ≼ 𝑦)
56	12, 30	unex 7689	. . . . . . 7 ⊢ (𝑦 ∪ 𝑥) ∈ V
57		ssun1 4129	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝑥)
58		ssdomg 8939	. . . . . . 7 ⊢ ((𝑦 ∪ 𝑥) ∈ V → (𝑦 ⊆ (𝑦 ∪ 𝑥) → 𝑦 ≼ (𝑦 ∪ 𝑥)))
59	56, 57, 58	mp2 9	. . . . . 6 ⊢ 𝑦 ≼ (𝑦 ∪ 𝑥)
60		sbth 9027	. . . . . 6 ⊢ (((𝑦 ∪ 𝑥) ≼ 𝑦 ∧ 𝑦 ≼ (𝑦 ∪ 𝑥)) → (𝑦 ∪ 𝑥) ≈ 𝑦)
61	55, 59, 60	sylancl 587	. . . . 5 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ≈ 𝑦)
62	8, 28	eqtr4d 2773	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = (♯‘𝑥))
63		hashen 14272	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ 𝑥 ∈ Fin) → ((♯‘𝑦) = (♯‘𝑥) ↔ 𝑦 ≈ 𝑥))
64	15, 33, 63	syl2anc 585	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → ((♯‘𝑦) = (♯‘𝑥) ↔ 𝑦 ≈ 𝑥))
65	62, 64	mpbid 232	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ≈ 𝑥)
66		fiuneneq 43471	. . . . . 6 ⊢ ((𝑦 ≈ 𝑥 ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ 𝑥) ≈ 𝑦 ↔ 𝑦 = 𝑥))
67	65, 15, 66	syl2anc 585	. . . . 5 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → ((𝑦 ∪ 𝑥) ≈ 𝑦 ↔ 𝑦 = 𝑥))
68	61, 67	mpbid 232	. . . 4 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 = 𝑥)
69	68	3expia 1122	. . 3 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺))) → (((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
70	69	ralrimivva 3178	. 2 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
71		fveqeq2 6842	. . 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 234	1 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∃*𝑦 ∈ (SubGrp‘𝐺)(♯‘𝑦) = 𝑁)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃*wrmo 3348  {crab 3398  Vcvv 3439   ∪ cun 3898   ⊆ wss 3900   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358   ≈ cen 8882   ≼ cdom 8883  Fincfn 8885   ≤ cle 11169  ℕcn 12147  ℕ₀cn0 12403  ♯chash 14255   ∥ cdvds 16181  Basecbs 17138   ↾_s cress 17159  SubGrpcsubg 19052  odcod 19455  mulGrpcmgp 20077  Unitcui 20293  IDomncidom 20628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106  ax-addf 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-disj 5065  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-omul 8402  df-er 8635  df-ec 8637  df-qs 8641  df-map 8767  df-pm 8768  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fsupp 9267  df-sup 9347  df-inf 9348  df-oi 9417  df-dju 9815  df-card 9853  df-acn 9856  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-xnn0 12477  df-z 12491  df-dec 12610  df-uz 12754  df-rp 12908  df-fz 13426  df-fzo 13573  df-fl 13714  df-mod 13792  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-sum 15612  df-dvds 16182  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-starv 17194  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-unif 17202  df-hom 17203  df-cco 17204  df-0g 17363  df-gsum 17364  df-prds 17369  df-pws 17371  df-mre 17507  df-mrc 17508  df-acs 17510  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-eqg 19057  df-ghm 19144  df-cntz 19248  df-od 19459  df-cmn 19713  df-abl 19714  df-mgp 20078  df-rng 20090  df-ur 20119  df-srg 20124  df-ring 20172  df-cring 20173  df-oppr 20275  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-rhm 20410  df-nzr 20448  df-subrng 20481  df-subrg 20505  df-rlreg 20629  df-domn 20630  df-idom 20631  df-lmod 20815  df-lss 20885  df-lsp 20925  df-cnfld 21312  df-assa 21810  df-asp 21811  df-ascl 21812  df-psr 21867  df-mvr 21868  df-mpl 21869  df-opsr 21871  df-evls 22031  df-evl 22032  df-psr1 22122  df-vr1 22123  df-ply1 22124  df-coe1 22125  df-evl1 22262  df-mdeg 26018  df-deg1 26019  df-mon1 26094  df-uc1p 26095  df-q1p 26096  df-r1p 26097

This theorem is referenced by:  proot1mul  43473