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Theorem idomsubgmo 42028
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 = ((mulGrpβ€˜π‘…) β†Ύs (Unitβ€˜π‘…))
Assertion
Ref Expression
idomsubgmo ((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) β†’ βˆƒ*𝑦 ∈ (SubGrpβ€˜πΊ)(β™―β€˜π‘¦) = 𝑁)
Distinct variable groups:   𝑦,𝐺   𝑦,𝑁   𝑦,𝑅

Proof of Theorem idomsubgmo
Dummy variables π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6904 . . . . . . . . 9 (Baseβ€˜πΊ) ∈ V
21rabex 5332 . . . . . . . 8 {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ V
3 simp2l 1199 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑦 ∈ (SubGrpβ€˜πΊ))
4 eqid 2732 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
54subgss 19009 . . . . . . . . . . 11 (𝑦 ∈ (SubGrpβ€˜πΊ) β†’ 𝑦 βŠ† (Baseβ€˜πΊ))
63, 5syl 17 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑦 βŠ† (Baseβ€˜πΊ))
7 simpl2l 1226 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ 𝑦) β†’ 𝑦 ∈ (SubGrpβ€˜πΊ))
8 simp3l 1201 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (β™―β€˜π‘¦) = 𝑁)
9 simp1r 1198 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑁 ∈ β„•)
109nnnn0d 12534 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑁 ∈ β„•0)
118, 10eqeltrd 2833 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (β™―β€˜π‘¦) ∈ β„•0)
12 vex 3478 . . . . . . . . . . . . . . 15 𝑦 ∈ V
13 hashclb 14320 . . . . . . . . . . . . . . 15 (𝑦 ∈ V β†’ (𝑦 ∈ Fin ↔ (β™―β€˜π‘¦) ∈ β„•0))
1412, 13ax-mp 5 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin ↔ (β™―β€˜π‘¦) ∈ β„•0)
1511, 14sylibr 233 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑦 ∈ Fin)
1615adantr 481 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ 𝑦) β†’ 𝑦 ∈ Fin)
17 simpr 485 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ 𝑦) β†’ 𝑧 ∈ 𝑦)
18 eqid 2732 . . . . . . . . . . . . 13 (odβ€˜πΊ) = (odβ€˜πΊ)
1918odsubdvds 19441 . . . . . . . . . . . 12 ((𝑦 ∈ (SubGrpβ€˜πΊ) ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ 𝑦) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ (β™―β€˜π‘¦))
207, 16, 17, 19syl3anc 1371 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ 𝑦) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ (β™―β€˜π‘¦))
218adantr 481 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ 𝑦) β†’ (β™―β€˜π‘¦) = 𝑁)
2220, 21breqtrd 5174 . . . . . . . . . 10 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ 𝑦) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁)
236, 22ssrabdv 4071 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑦 βŠ† {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁})
24 simp2r 1200 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ π‘₯ ∈ (SubGrpβ€˜πΊ))
254subgss 19009 . . . . . . . . . . 11 (π‘₯ ∈ (SubGrpβ€˜πΊ) β†’ π‘₯ βŠ† (Baseβ€˜πΊ))
2624, 25syl 17 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ π‘₯ βŠ† (Baseβ€˜πΊ))
27 simpl2r 1227 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ π‘₯) β†’ π‘₯ ∈ (SubGrpβ€˜πΊ))
28 simp3r 1202 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (β™―β€˜π‘₯) = 𝑁)
2928, 10eqeltrd 2833 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (β™―β€˜π‘₯) ∈ β„•0)
30 vex 3478 . . . . . . . . . . . . . . 15 π‘₯ ∈ V
31 hashclb 14320 . . . . . . . . . . . . . . 15 (π‘₯ ∈ V β†’ (π‘₯ ∈ Fin ↔ (β™―β€˜π‘₯) ∈ β„•0))
3230, 31ax-mp 5 . . . . . . . . . . . . . 14 (π‘₯ ∈ Fin ↔ (β™―β€˜π‘₯) ∈ β„•0)
3329, 32sylibr 233 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ π‘₯ ∈ Fin)
3433adantr 481 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ π‘₯) β†’ π‘₯ ∈ Fin)
35 simpr 485 . . . . . . . . . . . 12 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ π‘₯) β†’ 𝑧 ∈ π‘₯)
3618odsubdvds 19441 . . . . . . . . . . . 12 ((π‘₯ ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ Fin ∧ 𝑧 ∈ π‘₯) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ (β™―β€˜π‘₯))
3727, 34, 35, 36syl3anc 1371 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ π‘₯) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ (β™―β€˜π‘₯))
3828adantr 481 . . . . . . . . . . 11 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ π‘₯) β†’ (β™―β€˜π‘₯) = 𝑁)
3937, 38breqtrd 5174 . . . . . . . . . 10 ((((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) ∧ 𝑧 ∈ π‘₯) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁)
4026, 39ssrabdv 4071 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ π‘₯ βŠ† {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁})
4123, 40unssd 4186 . . . . . . . 8 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (𝑦 βˆͺ π‘₯) βŠ† {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁})
42 ssdomg 8998 . . . . . . . 8 ({𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ V β†’ ((𝑦 βˆͺ π‘₯) βŠ† {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} β†’ (𝑦 βˆͺ π‘₯) β‰Ό {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}))
432, 41, 42mpsyl 68 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (𝑦 βˆͺ π‘₯) β‰Ό {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁})
44 idomsubgmo.g . . . . . . . . . . 11 𝐺 = ((mulGrpβ€˜π‘…) β†Ύs (Unitβ€˜π‘…))
4544, 4, 18idomodle 42026 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜{𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}) ≀ 𝑁)
46453ad2ant1 1133 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (β™―β€˜{𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}) ≀ 𝑁)
4746, 8breqtrrd 5176 . . . . . . . 8 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (β™―β€˜{𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}) ≀ (β™―β€˜π‘¦))
482a1i 11 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ V)
49 hashbnd 14298 . . . . . . . . . 10 (({𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ V ∧ (β™―β€˜π‘¦) ∈ β„•0 ∧ (β™―β€˜{𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}) ≀ (β™―β€˜π‘¦)) β†’ {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ Fin)
5048, 11, 47, 49syl3anc 1371 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ Fin)
51 hashdom 14341 . . . . . . . . 9 (({𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ Fin ∧ 𝑦 ∈ V) β†’ ((β™―β€˜{𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}) ≀ (β™―β€˜π‘¦) ↔ {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} β‰Ό 𝑦))
5250, 12, 51sylancl 586 . . . . . . . 8 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ ((β™―β€˜{𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}) ≀ (β™―β€˜π‘¦) ↔ {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} β‰Ό 𝑦))
5347, 52mpbid 231 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} β‰Ό 𝑦)
54 domtr 9005 . . . . . . 7 (((𝑦 βˆͺ π‘₯) β‰Ό {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∧ {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} β‰Ό 𝑦) β†’ (𝑦 βˆͺ π‘₯) β‰Ό 𝑦)
5543, 53, 54syl2anc 584 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (𝑦 βˆͺ π‘₯) β‰Ό 𝑦)
5612, 30unex 7735 . . . . . . 7 (𝑦 βˆͺ π‘₯) ∈ V
57 ssun1 4172 . . . . . . 7 𝑦 βŠ† (𝑦 βˆͺ π‘₯)
58 ssdomg 8998 . . . . . . 7 ((𝑦 βˆͺ π‘₯) ∈ V β†’ (𝑦 βŠ† (𝑦 βˆͺ π‘₯) β†’ 𝑦 β‰Ό (𝑦 βˆͺ π‘₯)))
5956, 57, 58mp2 9 . . . . . 6 𝑦 β‰Ό (𝑦 βˆͺ π‘₯)
60 sbth 9095 . . . . . 6 (((𝑦 βˆͺ π‘₯) β‰Ό 𝑦 ∧ 𝑦 β‰Ό (𝑦 βˆͺ π‘₯)) β†’ (𝑦 βˆͺ π‘₯) β‰ˆ 𝑦)
6155, 59, 60sylancl 586 . . . . 5 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (𝑦 βˆͺ π‘₯) β‰ˆ 𝑦)
628, 28eqtr4d 2775 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ (β™―β€˜π‘¦) = (β™―β€˜π‘₯))
63 hashen 14309 . . . . . . . 8 ((𝑦 ∈ Fin ∧ π‘₯ ∈ Fin) β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘₯) ↔ 𝑦 β‰ˆ π‘₯))
6415, 33, 63syl2anc 584 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘₯) ↔ 𝑦 β‰ˆ π‘₯))
6562, 64mpbid 231 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑦 β‰ˆ π‘₯)
66 fiuneneq 42027 . . . . . 6 ((𝑦 β‰ˆ π‘₯ ∧ 𝑦 ∈ Fin) β†’ ((𝑦 βˆͺ π‘₯) β‰ˆ 𝑦 ↔ 𝑦 = π‘₯))
6765, 15, 66syl2anc 584 . . . . 5 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ ((𝑦 βˆͺ π‘₯) β‰ˆ 𝑦 ↔ 𝑦 = π‘₯))
6861, 67mpbid 231 . . . 4 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁)) β†’ 𝑦 = π‘₯)
69683expia 1121 . . 3 (((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) ∧ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ (SubGrpβ€˜πΊ))) β†’ (((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁) β†’ 𝑦 = π‘₯))
7069ralrimivva 3200 . 2 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) β†’ βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)βˆ€π‘₯ ∈ (SubGrpβ€˜πΊ)(((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁) β†’ 𝑦 = π‘₯))
71 fveqeq2 6900 . . 3 (𝑦 = π‘₯ β†’ ((β™―β€˜π‘¦) = 𝑁 ↔ (β™―β€˜π‘₯) = 𝑁))
7271rmo4 3726 . 2 (βˆƒ*𝑦 ∈ (SubGrpβ€˜πΊ)(β™―β€˜π‘¦) = 𝑁 ↔ βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)βˆ€π‘₯ ∈ (SubGrpβ€˜πΊ)(((β™―β€˜π‘¦) = 𝑁 ∧ (β™―β€˜π‘₯) = 𝑁) β†’ 𝑦 = π‘₯))
7370, 72sylibr 233 1 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) β†’ βˆƒ*𝑦 ∈ (SubGrpβ€˜πΊ)(β™―β€˜π‘¦) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒ*wrmo 3375  {crab 3432  Vcvv 3474   βˆͺ cun 3946   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411   β‰ˆ cen 8938   β‰Ό cdom 8939  Fincfn 8941   ≀ cle 11251  β„•cn 12214  β„•0cn0 12474  β™―chash 14292   βˆ₯ cdvds 16199  Basecbs 17146   β†Ύs cress 17175  SubGrpcsubg 19002  odcod 19394  mulGrpcmgp 19989  Unitcui 20173  IDomncidom 20903
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-ofr 7673  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-acn 9939  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-xnn0 12547  df-z 12561  df-dec 12680  df-uz 12825  df-rp 12977  df-fz 13487  df-fzo 13630  df-fl 13759  df-mod 13837  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-sum 15635  df-dvds 16200  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-mulr 17213  df-starv 17214  df-sca 17215  df-vsca 17216  df-ip 17217  df-tset 17218  df-ple 17219  df-ds 17221  df-unif 17222  df-hom 17223  df-cco 17224  df-0g 17389  df-gsum 17390  df-prds 17395  df-pws 17397  df-mre 17532  df-mrc 17533  df-acs 17535  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-submnd 18674  df-grp 18824  df-minusg 18825  df-sbg 18826  df-mulg 18953  df-subg 19005  df-eqg 19007  df-ghm 19092  df-cntz 19183  df-od 19398  df-cmn 19652  df-abl 19653  df-mgp 19990  df-ur 20007  df-srg 20012  df-ring 20060  df-cring 20061  df-oppr 20154  df-dvdsr 20175  df-unit 20176  df-invr 20206  df-rnghom 20255  df-nzr 20296  df-subrg 20321  df-lmod 20477  df-lss 20548  df-lsp 20588  df-rlreg 20905  df-domn 20906  df-idom 20907  df-cnfld 20951  df-assa 21414  df-asp 21415  df-ascl 21416  df-psr 21468  df-mvr 21469  df-mpl 21470  df-opsr 21472  df-evls 21641  df-evl 21642  df-psr1 21710  df-vr1 21711  df-ply1 21712  df-coe1 21713  df-evl1 21842  df-mdeg 25577  df-deg1 25578  df-mon1 25655  df-uc1p 25656  df-q1p 25657  df-r1p 25658
This theorem is referenced by:  proot1mul  42029
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