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Theorem idomsubgmo 37296
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.
StepHypRef Expression
1 fvex 6168 . . . . . . . . 9 (Base‘𝐺) ∈ V
21rabex 4783 . . . . . . . 8 {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V
3 simp2l 1085 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑦 ∈ (SubGrp‘𝐺))
4 eqid 2621 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
54subgss 17535 . . . . . . . . . . 11 (𝑦 ∈ (SubGrp‘𝐺) → 𝑦 ⊆ (Base‘𝐺))
63, 5syl 17 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑦 ⊆ (Base‘𝐺))
7 simpl2l 1112 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → 𝑦 ∈ (SubGrp‘𝐺))
8 simp3l 1087 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (#‘𝑦) = 𝑁)
9 simp1r 1084 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ)
109nnnn0d 11311 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ0)
118, 10eqeltrd 2698 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (#‘𝑦) ∈ ℕ0)
12 vex 3193 . . . . . . . . . . . . . . 15 𝑦 ∈ V
13 hashclb 13105 . . . . . . . . . . . . . . 15 (𝑦 ∈ V → (𝑦 ∈ Fin ↔ (#‘𝑦) ∈ ℕ0))
1412, 13ax-mp 5 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin ↔ (#‘𝑦) ∈ ℕ0)
1511, 14sylibr 224 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑦 ∈ Fin)
1615adantr 481 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → 𝑦 ∈ Fin)
17 simpr 477 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → 𝑧𝑦)
18 eqid 2621 . . . . . . . . . . . . 13 (od‘𝐺) = (od‘𝐺)
1918odsubdvds 17926 . . . . . . . . . . . 12 ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ Fin ∧ 𝑧𝑦) → ((od‘𝐺)‘𝑧) ∥ (#‘𝑦))
207, 16, 17, 19syl3anc 1323 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → ((od‘𝐺)‘𝑧) ∥ (#‘𝑦))
218adantr 481 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → (#‘𝑦) = 𝑁)
2220, 21breqtrd 4649 . . . . . . . . . 10 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → ((od‘𝐺)‘𝑧) ∥ 𝑁)
236, 22ssrabdv 3666 . . . . . . . . 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 1086 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑥 ∈ (SubGrp‘𝐺))
254subgss 17535 . . . . . . . . . . 11 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
2624, 25syl 17 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑥 ⊆ (Base‘𝐺))
27 simpl2r 1113 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → 𝑥 ∈ (SubGrp‘𝐺))
28 simp3r 1088 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (#‘𝑥) = 𝑁)
2928, 10eqeltrd 2698 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (#‘𝑥) ∈ ℕ0)
30 vex 3193 . . . . . . . . . . . . . . 15 𝑥 ∈ V
31 hashclb 13105 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → (𝑥 ∈ Fin ↔ (#‘𝑥) ∈ ℕ0))
3230, 31ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∈ Fin ↔ (#‘𝑥) ∈ ℕ0)
3329, 32sylibr 224 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑥 ∈ Fin)
3433adantr 481 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → 𝑥 ∈ Fin)
35 simpr 477 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → 𝑧𝑥)
3618odsubdvds 17926 . . . . . . . . . . . 12 ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ Fin ∧ 𝑧𝑥) → ((od‘𝐺)‘𝑧) ∥ (#‘𝑥))
3727, 34, 35, 36syl3anc 1323 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → ((od‘𝐺)‘𝑧) ∥ (#‘𝑥))
3828adantr 481 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → (#‘𝑥) = 𝑁)
3937, 38breqtrd 4649 . . . . . . . . . 10 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → ((od‘𝐺)‘𝑧) ∥ 𝑁)
4026, 39ssrabdv 3666 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑥 ⊆ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁})
4123, 40unssd 3773 . . . . . . . 8 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (𝑦𝑥) ⊆ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁})
42 ssdomg 7961 . . . . . . . 8 ({𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V → ((𝑦𝑥) ⊆ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} → (𝑦𝑥) ≼ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}))
432, 41, 42mpsyl 68 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (𝑦𝑥) ≼ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁})
44 idomsubgmo.g . . . . . . . . . . 11 𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))
4544, 4, 18idomodle 37294 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → (#‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ 𝑁)
46453ad2ant1 1080 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (#‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ 𝑁)
4746, 8breqtrrd 4651 . . . . . . . 8 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (#‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ (#‘𝑦))
482a1i 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 13079 . . . . . . . . . 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)
5048, 11, 47, 49syl3anc 1323 . . . . . . . . 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 13124 . . . . . . . . 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))
5250, 12, 51sylancl 693 . . . . . . . 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))
5347, 52mpbid 222 . . . . . . 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 7969 . . . . . . 7 (((𝑦𝑥) ≼ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∧ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ≼ 𝑦) → (𝑦𝑥) ≼ 𝑦)
5543, 53, 54syl2anc 692 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (𝑦𝑥) ≼ 𝑦)
5612, 30unex 6921 . . . . . . 7 (𝑦𝑥) ∈ V
57 ssun1 3760 . . . . . . 7 𝑦 ⊆ (𝑦𝑥)
58 ssdomg 7961 . . . . . . 7 ((𝑦𝑥) ∈ V → (𝑦 ⊆ (𝑦𝑥) → 𝑦 ≼ (𝑦𝑥)))
5956, 57, 58mp2 9 . . . . . 6 𝑦 ≼ (𝑦𝑥)
60 sbth 8040 . . . . . 6 (((𝑦𝑥) ≼ 𝑦𝑦 ≼ (𝑦𝑥)) → (𝑦𝑥) ≈ 𝑦)
6155, 59, 60sylancl 693 . . . . 5 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (𝑦𝑥) ≈ 𝑦)
628, 28eqtr4d 2658 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → (#‘𝑦) = (#‘𝑥))
63 hashen 13091 . . . . . . . 8 ((𝑦 ∈ Fin ∧ 𝑥 ∈ Fin) → ((#‘𝑦) = (#‘𝑥) ↔ 𝑦𝑥))
6415, 33, 63syl2anc 692 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → ((#‘𝑦) = (#‘𝑥) ↔ 𝑦𝑥))
6562, 64mpbid 222 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑦𝑥)
66 fiuneneq 37295 . . . . . 6 ((𝑦𝑥𝑦 ∈ Fin) → ((𝑦𝑥) ≈ 𝑦𝑦 = 𝑥))
6765, 15, 66syl2anc 692 . . . . 5 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → ((𝑦𝑥) ≈ 𝑦𝑦 = 𝑥))
6861, 67mpbid 222 . . . 4 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁)) → 𝑦 = 𝑥)
69683expia 1264 . . 3 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺))) → (((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁) → 𝑦 = 𝑥))
7069ralrimivva 2967 . 2 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁) → 𝑦 = 𝑥))
71 fveq2 6158 . . . 4 (𝑦 = 𝑥 → (#‘𝑦) = (#‘𝑥))
7271eqeq1d 2623 . . 3 (𝑦 = 𝑥 → ((#‘𝑦) = 𝑁 ↔ (#‘𝑥) = 𝑁))
7372rmo4 3386 . 2 (∃*𝑦 ∈ (SubGrp‘𝐺)(#‘𝑦) = 𝑁 ↔ ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((#‘𝑦) = 𝑁 ∧ (#‘𝑥) = 𝑁) → 𝑦 = 𝑥))
7470, 73sylibr 224 1 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∃*𝑦 ∈ (SubGrp‘𝐺)(#‘𝑦) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  ∃*wrmo 2911  {crab 2912  Vcvv 3190  cun 3558  wss 3560   class class class wbr 4623  cfv 5857  (class class class)co 6615  cen 7912  cdom 7913  Fincfn 7915  cle 10035  cn 10980  0cn0 11252  #chash 13073  cdvds 14926  Basecbs 15800  s cress 15801  SubGrpcsubg 17528  odcod 17884  mulGrpcmgp 18429  Unitcui 18579  IDomncidom 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975  ax-mulf 9976
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-disj 4594  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-tpos 7312  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-omul 7525  df-er 7702  df-ec 7704  df-qs 7708  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-acn 8728  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-xnn0 11324  df-z 11338  df-dec 11454  df-uz 11648  df-rp 11793  df-fz 12285  df-fzo 12423  df-fl 12549  df-mod 12625  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-sum 14367  df-dvds 14927  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-starv 15896  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-unif 15905  df-hom 15906  df-cco 15907  df-0g 16042  df-gsum 16043  df-prds 16048  df-pws 16050  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-submnd 17276  df-grp 17365  df-minusg 17366  df-sbg 17367  df-mulg 17481  df-subg 17531  df-eqg 17533  df-ghm 17598  df-cntz 17690  df-od 17888  df-cmn 18135  df-abl 18136  df-mgp 18430  df-ur 18442  df-srg 18446  df-ring 18489  df-cring 18490  df-oppr 18563  df-dvdsr 18581  df-unit 18582  df-invr 18612  df-rnghom 18655  df-subrg 18718  df-lmod 18805  df-lss 18873  df-lsp 18912  df-nzr 19198  df-rlreg 19223  df-domn 19224  df-idom 19225  df-assa 19252  df-asp 19253  df-ascl 19254  df-psr 19296  df-mvr 19297  df-mpl 19298  df-opsr 19300  df-evls 19446  df-evl 19447  df-psr1 19490  df-vr1 19491  df-ply1 19492  df-coe1 19493  df-evl1 19621  df-cnfld 19687  df-mdeg 23753  df-deg1 23754  df-mon1 23828  df-uc1p 23829  df-q1p 23830  df-r1p 23831
This theorem is referenced by:  proot1mul  37297
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