Theorem idomsubgmo 43205

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 6919	. . . . . . . . 9 ⊢ (Base‘𝐺) ∈ V
2	1	rabex 5339	. . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19145	. . . . . . . . . . 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 12587	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ0)
11	8, 10	eqeltrd 2841	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) ∈ ℕ0)
12		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
13		hashclb 14397	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (od‘𝐺) = (od‘𝐺)
19	18	odsubdvds 19589	. . . . . . . . . . . 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 5169	. . . . . . . . . 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 4074	. . . . . . . . 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 19145	. . . . . . . . . . 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 2841	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) ∈ ℕ0)
30		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
31		hashclb 14397	. . . . . . . . . . . . . . 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 19589	. . . . . . . . . . . 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 5169	. . . . . . . . . 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 4074	. . . . . . . . 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 4192	. . . . . . . 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 9040	. . . . . . . 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 43203	. . . . . . . . . 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 5171	. . . . . . . 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 14375	. . . . . . . . . 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 14418	. . . . . . . . 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 9047	. . . . . . 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 7764	. . . . . . 7 ⊢ (𝑦 ∪ 𝑥) ∈ V
57		ssun1 4178	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝑥)
58		ssdomg 9040	. . . . . . 7 ⊢ ((𝑦 ∪ 𝑥) ∈ V → (𝑦 ⊆ (𝑦 ∪ 𝑥) → 𝑦 ≼ (𝑦 ∪ 𝑥)))
59	56, 57, 58	mp2 9	. . . . . 6 ⊢ 𝑦 ≼ (𝑦 ∪ 𝑥)
60		sbth 9133	. . . . . 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 2780	. . . . . . 7 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = (♯‘𝑥))
63		hashen 14386	. . . . . . . 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 43204	. . . . . 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 1122	. . 3 ⊢ (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺))) → (((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
70	69	ralrimivva 3202	. 2 ⊢ ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
71		fveqeq2 6915	. . 3 ⊢ (𝑦 = 𝑥 → ((♯‘𝑦) = 𝑁 ↔ (♯‘𝑥) = 𝑁))
72	71	rmo4 3736	. 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 1540   ∈ wcel 2108  ∀wral 3061  ∃*wrmo 3379  {crab 3436  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431   ≈ cen 8982   ≼ cdom 8983  Fincfn 8985   ≤ cle 11296  ℕcn 12266  ℕ₀cn0 12526  ♯chash 14369   ∥ cdvds 16290  Basecbs 17247   ↾_s cress 17274  SubGrpcsubg 19138  odcod 19542  mulGrpcmgp 20137  Unitcui 20355  IDomncidom 20693

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-eqg 19143  df-ghm 19231  df-cntz 19335  df-od 19546  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-rhm 20472  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-domn 20695  df-idom 20696  df-lmod 20860  df-lss 20930  df-lsp 20970  df-cnfld 21365  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evl1 22320  df-mdeg 26094  df-deg1 26095  df-mon1 26170  df-uc1p 26171  df-q1p 26172  df-r1p 26173

This theorem is referenced by:  proot1mul  43206