Theorem idomsubgmo 43175

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 6853	. . . . . . . . 9 ⊢ (Base‘𝐺) ∈ V
2	1	rabex 5289	. . . . . . . 8 ⊢ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V
3		simp2l 1200	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ∈ (SubGrp‘𝐺))
4		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19041	. . . . . . . . . . 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 1227	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑦) → 𝑦 ∈ (SubGrp‘𝐺))
8		simp3l 1202	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = 𝑁)
9		simp1r 1199	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ)
10	9	nnnn0d 12479	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ0)
11	8, 10	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) ∈ ℕ0)
12		vex 3448	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
13		hashclb 14299	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (od‘𝐺) = (od‘𝐺)
19	18	odsubdvds 19485	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ 𝑦) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑦))
20	7, 16, 17, 19	syl3anc 1373	. . . . . . . . . . 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 5128	. . . . . . . . . 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 4033	. . . . . . . . 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 1201	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ∈ (SubGrp‘𝐺))
25	4	subgss 19041	. . . . . . . . . . 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 1228	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (SubGrp‘𝐺))
28		simp3r 1203	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) = 𝑁)
29	28, 10	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) ∈ ℕ0)
30		vex 3448	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
31		hashclb 14299	. . . . . . . . . . . . . . 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 19485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ Fin ∧ 𝑧 ∈ 𝑥) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑥))
37	27, 34, 35, 36	syl3anc 1373	. . . . . . . . . . 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 5128	. . . . . . . . . 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 4033	. . . . . . . . 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 4151	. . . . . . . 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 8948	. . . . . . . 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 43173	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → (♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ 𝑁)
46	45	3ad2ant1 1133	. . . . . . . . 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 5130	. . . . . . . 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 14277	. . . . . . . . . 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 1373	. . . . . . . . 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 14320	. . . . . . . . 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 586	. . . . . . . 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 8955	. . . . . . 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 584	. . . . . 6 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ≼ 𝑦)
56	12, 30	unex 7700	. . . . . . 7 ⊢ (𝑦 ∪ 𝑥) ∈ V
57		ssun1 4137	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝑥)
58		ssdomg 8948	. . . . . . 7 ⊢ ((𝑦 ∪ 𝑥) ∈ V → (𝑦 ⊆ (𝑦 ∪ 𝑥) → 𝑦 ≼ (𝑦 ∪ 𝑥)))
59	56, 57, 58	mp2 9	. . . . . 6 ⊢ 𝑦 ≼ (𝑦 ∪ 𝑥)
60		sbth 9038	. . . . . 6 ⊢ (((𝑦 ∪ 𝑥) ≼ 𝑦 ∧ 𝑦 ≼ (𝑦 ∪ 𝑥)) → (𝑦 ∪ 𝑥) ≈ 𝑦)
61	55, 59, 60	sylancl 586	. . . . 5 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦 ∪ 𝑥) ≈ 𝑦)
62	8, 28	eqtr4d 2767	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = (♯‘𝑥))
63		hashen 14288	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ 𝑥 ∈ Fin) → ((♯‘𝑦) = (♯‘𝑥) ↔ 𝑦 ≈ 𝑥))
64	15, 33, 63	syl2anc 584	. . . . . . 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 43174	. . . . . 6 ⊢ ((𝑦 ≈ 𝑥 ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ 𝑥) ≈ 𝑦 ↔ 𝑦 = 𝑥))
67	65, 15, 66	syl2anc 584	. . . . 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 1121	. . 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 6849	. . 3 ⊢ (𝑦 = 𝑥 → ((♯‘𝑦) = 𝑁 ↔ (♯‘𝑥) = 𝑁))
72	71	rmo4 3698	. 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃*wrmo 3350  {crab 3402  Vcvv 3444   ∪ cun 3909   ⊆ wss 3911   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369   ≈ cen 8892   ≼ cdom 8893  Fincfn 8895   ≤ cle 11185  ℕcn 12162  ℕ₀cn0 12418  ♯chash 14271   ∥ cdvds 16198  Basecbs 17155   ↾_s cress 17176  SubGrpcsubg 19034  odcod 19438  mulGrpcmgp 20060  Unitcui 20275  IDomncidom 20613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-disj 5070  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-omul 8416  df-er 8648  df-ec 8650  df-qs 8654  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-inf 9370  df-oi 9439  df-dju 9830  df-card 9868  df-acn 9871  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-rp 12928  df-fz 13445  df-fzo 13592  df-fl 13730  df-mod 13808  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-sum 15629  df-dvds 16199  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-0g 17380  df-gsum 17381  df-prds 17386  df-pws 17388  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-submnd 18693  df-grp 18850  df-minusg 18851  df-sbg 18852  df-mulg 18982  df-subg 19037  df-eqg 19039  df-ghm 19127  df-cntz 19231  df-od 19442  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-srg 20107  df-ring 20155  df-cring 20156  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-invr 20308  df-rhm 20392  df-nzr 20433  df-subrng 20466  df-subrg 20490  df-rlreg 20614  df-domn 20615  df-idom 20616  df-lmod 20800  df-lss 20870  df-lsp 20910  df-cnfld 21297  df-assa 21795  df-asp 21796  df-ascl 21797  df-psr 21851  df-mvr 21852  df-mpl 21853  df-opsr 21855  df-evls 22014  df-evl 22015  df-psr1 22097  df-vr1 22098  df-ply1 22099  df-coe1 22100  df-evl1 22236  df-mdeg 25993  df-deg1 25994  df-mon1 26069  df-uc1p 26070  df-q1p 26071  df-r1p 26072

This theorem is referenced by:  proot1mul  43176