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Theorem idomsubgmo 39194
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 6514 . . . . . . . . 9 (Base‘𝐺) ∈ V
21rabex 5092 . . . . . . . 8 {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∈ V
3 simp2l 1179 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ∈ (SubGrp‘𝐺))
4 eqid 2778 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
54subgss 18067 . . . . . . . . . . 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 1206 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → 𝑦 ∈ (SubGrp‘𝐺))
8 simp3l 1181 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = 𝑁)
9 simp1r 1178 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ)
109nnnn0d 11770 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑁 ∈ ℕ0)
118, 10eqeltrd 2866 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) ∈ ℕ0)
12 vex 3418 . . . . . . . . . . . . . . 15 𝑦 ∈ V
13 hashclb 13537 . . . . . . . . . . . . . . 15 (𝑦 ∈ V → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ0))
1412, 13ax-mp 5 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ0)
1511, 14sylibr 226 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 ∈ Fin)
1615adantr 473 . . . . . . . . . . . 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 2778 . . . . . . . . . . . . 13 (od‘𝐺) = (od‘𝐺)
1918odsubdvds 18460 . . . . . . . . . . . 12 ((𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ Fin ∧ 𝑧𝑦) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑦))
207, 16, 17, 19syl3anc 1351 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑦))
218adantr 473 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → (♯‘𝑦) = 𝑁)
2220, 21breqtrd 4956 . . . . . . . . . 10 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑦) → ((od‘𝐺)‘𝑧) ∥ 𝑁)
236, 22ssrabdv 3942 . . . . . . . . 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 1180 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ∈ (SubGrp‘𝐺))
254subgss 18067 . . . . . . . . . . 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 1207 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → 𝑥 ∈ (SubGrp‘𝐺))
28 simp3r 1182 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) = 𝑁)
2928, 10eqeltrd 2866 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑥) ∈ ℕ0)
30 vex 3418 . . . . . . . . . . . . . . 15 𝑥 ∈ V
31 hashclb 13537 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → (𝑥 ∈ Fin ↔ (♯‘𝑥) ∈ ℕ0))
3230, 31ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∈ Fin ↔ (♯‘𝑥) ∈ ℕ0)
3329, 32sylibr 226 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑥 ∈ Fin)
3433adantr 473 . . . . . . . . . . . 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 18460 . . . . . . . . . . . 12 ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ Fin ∧ 𝑧𝑥) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑥))
3727, 34, 35, 36syl3anc 1351 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → ((od‘𝐺)‘𝑧) ∥ (♯‘𝑥))
3828adantr 473 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → (♯‘𝑥) = 𝑁)
3937, 38breqtrd 4956 . . . . . . . . . 10 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) ∧ 𝑧𝑥) → ((od‘𝐺)‘𝑧) ∥ 𝑁)
4026, 39ssrabdv 3942 . . . . . . . . 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 4052 . . . . . . . 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 8354 . . . . . . . 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 39192 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → (♯‘{𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁}) ≤ 𝑁)
46453ad2ant1 1113 . . . . . . . . 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 4958 . . . . . . . 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 13514 . . . . . . . . . 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 1351 . . . . . . . . 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 13556 . . . . . . . . 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 577 . . . . . . . 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 224 . . . . . . 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 8361 . . . . . . 7 (((𝑦𝑥) ≼ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ∧ {𝑧 ∈ (Base‘𝐺) ∣ ((od‘𝐺)‘𝑧) ∥ 𝑁} ≼ 𝑦) → (𝑦𝑥) ≼ 𝑦)
5543, 53, 54syl2anc 576 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦𝑥) ≼ 𝑦)
5612, 30unex 7288 . . . . . . 7 (𝑦𝑥) ∈ V
57 ssun1 4039 . . . . . . 7 𝑦 ⊆ (𝑦𝑥)
58 ssdomg 8354 . . . . . . 7 ((𝑦𝑥) ∈ V → (𝑦 ⊆ (𝑦𝑥) → 𝑦 ≼ (𝑦𝑥)))
5956, 57, 58mp2 9 . . . . . 6 𝑦 ≼ (𝑦𝑥)
60 sbth 8435 . . . . . 6 (((𝑦𝑥) ≼ 𝑦𝑦 ≼ (𝑦𝑥)) → (𝑦𝑥) ≈ 𝑦)
6155, 59, 60sylancl 577 . . . . 5 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (𝑦𝑥) ≈ 𝑦)
628, 28eqtr4d 2817 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → (♯‘𝑦) = (♯‘𝑥))
63 hashen 13525 . . . . . . . 8 ((𝑦 ∈ Fin ∧ 𝑥 ∈ Fin) → ((♯‘𝑦) = (♯‘𝑥) ↔ 𝑦𝑥))
6415, 33, 63syl2anc 576 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → ((♯‘𝑦) = (♯‘𝑥) ↔ 𝑦𝑥))
6562, 64mpbid 224 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦𝑥)
66 fiuneneq 39193 . . . . . 6 ((𝑦𝑥𝑦 ∈ Fin) → ((𝑦𝑥) ≈ 𝑦𝑦 = 𝑥))
6765, 15, 66syl2anc 576 . . . . 5 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → ((𝑦𝑥) ≈ 𝑦𝑦 = 𝑥))
6861, 67mpbid 224 . . . 4 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ ((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁)) → 𝑦 = 𝑥)
69683expia 1101 . . 3 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺))) → (((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
7069ralrimivva 3141 . 2 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
71 fveqeq2 6510 . . 3 (𝑦 = 𝑥 → ((♯‘𝑦) = 𝑁 ↔ (♯‘𝑥) = 𝑁))
7271rmo4 3635 . 2 (∃*𝑦 ∈ (SubGrp‘𝐺)(♯‘𝑦) = 𝑁 ↔ ∀𝑦 ∈ (SubGrp‘𝐺)∀𝑥 ∈ (SubGrp‘𝐺)(((♯‘𝑦) = 𝑁 ∧ (♯‘𝑥) = 𝑁) → 𝑦 = 𝑥))
7370, 72sylibr 226 1 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∃*𝑦 ∈ (SubGrp‘𝐺)(♯‘𝑦) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3088  ∃*wrmo 3091  {crab 3092  Vcvv 3415  cun 3829  wss 3831   class class class wbr 4930  cfv 6190  (class class class)co 6978  cen 8305  cdom 8306  Fincfn 8308  cle 10477  cn 11441  0cn0 11710  chash 13508  cdvds 15470  Basecbs 16342  s cress 16343  SubGrpcsubg 18060  odcod 18417  mulGrpcmgp 18965  Unitcui 19115  IDomncidom 19778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-inf2 8900  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414  ax-pre-sup 10415  ax-addf 10416  ax-mulf 10417
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-iin 4796  df-disj 4899  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-se 5368  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-isom 6199  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-of 7229  df-ofr 7230  df-om 7399  df-1st 7503  df-2nd 7504  df-supp 7636  df-tpos 7697  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-2o 7908  df-oadd 7911  df-omul 7912  df-er 8091  df-ec 8093  df-qs 8097  df-map 8210  df-pm 8211  df-ixp 8262  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-fsupp 8631  df-sup 8703  df-inf 8704  df-oi 8771  df-dju 9126  df-card 9164  df-acn 9167  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-div 11101  df-nn 11442  df-2 11506  df-3 11507  df-4 11508  df-5 11509  df-6 11510  df-7 11511  df-8 11512  df-9 11513  df-n0 11711  df-xnn0 11783  df-z 11797  df-dec 11915  df-uz 12062  df-rp 12208  df-fz 12712  df-fzo 12853  df-fl 12980  df-mod 13056  df-seq 13188  df-exp 13248  df-hash 13509  df-cj 14322  df-re 14323  df-im 14324  df-sqrt 14458  df-abs 14459  df-clim 14709  df-sum 14907  df-dvds 15471  df-struct 16344  df-ndx 16345  df-slot 16346  df-base 16348  df-sets 16349  df-ress 16350  df-plusg 16437  df-mulr 16438  df-starv 16439  df-sca 16440  df-vsca 16441  df-ip 16442  df-tset 16443  df-ple 16444  df-ds 16446  df-unif 16447  df-hom 16448  df-cco 16449  df-0g 16574  df-gsum 16575  df-prds 16580  df-pws 16582  df-mre 16718  df-mrc 16719  df-acs 16721  df-mgm 17713  df-sgrp 17755  df-mnd 17766  df-mhm 17806  df-submnd 17807  df-grp 17897  df-minusg 17898  df-sbg 17899  df-mulg 18015  df-subg 18063  df-eqg 18065  df-ghm 18130  df-cntz 18221  df-od 18421  df-cmn 18671  df-abl 18672  df-mgp 18966  df-ur 18978  df-srg 18982  df-ring 19025  df-cring 19026  df-oppr 19099  df-dvdsr 19117  df-unit 19118  df-invr 19148  df-rnghom 19193  df-subrg 19259  df-lmod 19361  df-lss 19429  df-lsp 19469  df-nzr 19755  df-rlreg 19780  df-domn 19781  df-idom 19782  df-assa 19809  df-asp 19810  df-ascl 19811  df-psr 19853  df-mvr 19854  df-mpl 19855  df-opsr 19857  df-evls 20002  df-evl 20003  df-psr1 20054  df-vr1 20055  df-ply1 20056  df-coe1 20057  df-evl1 20185  df-cnfld 20251  df-mdeg 24355  df-deg1 24356  df-mon1 24430  df-uc1p 24431  df-q1p 24432  df-r1p 24433
This theorem is referenced by:  proot1mul  39195
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