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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idomsubgmo Structured version   Visualization version   GIF version

Theorem idomsubgmo 41568
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 6856 . . . . . . . . 9 (Baseβ€˜πΊ) ∈ V
21rabex 5290 . . . . . . . 8 {𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁} ∈ V
3 simp2l 1200 . . . . . . . . . . 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 2733 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
54subgss 18934 . . . . . . . . . . 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 1227 . . . . . . . . . . . 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 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)
9 simp1r 1199 . . . . . . . . . . . . . . . 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 12478 . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . 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 3448 . . . . . . . . . . . . . . 15 𝑦 ∈ V
13 hashclb 14264 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 486 . . . . . . . . . . . 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 2733 . . . . . . . . . . . . 13 (odβ€˜πΊ) = (odβ€˜πΊ)
1918odsubdvds 19358 . . . . . . . . . . . 12 ((𝑦 ∈ (SubGrpβ€˜πΊ) ∧ 𝑦 ∈ Fin ∧ 𝑧 ∈ 𝑦) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ (β™―β€˜π‘¦))
207, 16, 17, 19syl3anc 1372 . . . . . . . . . . 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 482 . . . . . . . . . . 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 5132 . . . . . . . . . 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 4032 . . . . . . . . 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 1201 . . . . . . . . . . 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 18934 . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 1203 . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . 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 3448 . . . . . . . . . . . . . . 15 π‘₯ ∈ V
31 hashclb 14264 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 486 . . . . . . . . . . . 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 19358 . . . . . . . . . . . 12 ((π‘₯ ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ Fin ∧ 𝑧 ∈ π‘₯) β†’ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ (β™―β€˜π‘₯))
3727, 34, 35, 36syl3anc 1372 . . . . . . . . . . 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 482 . . . . . . . . . . 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 5132 . . . . . . . . . 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 4032 . . . . . . . . 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 4147 . . . . . . . 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 8943 . . . . . . . 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 41566 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜{𝑧 ∈ (Baseβ€˜πΊ) ∣ ((odβ€˜πΊ)β€˜π‘§) βˆ₯ 𝑁}) ≀ 𝑁)
46453ad2ant1 1134 . . . . . . . . 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 5134 . . . . . . . 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 14242 . . . . . . . . . 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 1372 . . . . . . . . 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 14285 . . . . . . . . 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 587 . . . . . . . 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 8950 . . . . . . 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 585 . . . . . 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 7681 . . . . . . 7 (𝑦 βˆͺ π‘₯) ∈ V
57 ssun1 4133 . . . . . . 7 𝑦 βŠ† (𝑦 βˆͺ π‘₯)
58 ssdomg 8943 . . . . . . 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 9040 . . . . . 6 (((𝑦 βˆͺ π‘₯) β‰Ό 𝑦 ∧ 𝑦 β‰Ό (𝑦 βˆͺ π‘₯)) β†’ (𝑦 βˆͺ π‘₯) β‰ˆ 𝑦)
6155, 59, 60sylancl 587 . . . . 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 2776 . . . . . . 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 14253 . . . . . . . 8 ((𝑦 ∈ Fin ∧ π‘₯ ∈ Fin) β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘₯) ↔ 𝑦 β‰ˆ π‘₯))
6415, 33, 63syl2anc 585 . . . . . . 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 41567 . . . . . 6 ((𝑦 β‰ˆ π‘₯ ∧ 𝑦 ∈ Fin) β†’ ((𝑦 βˆͺ π‘₯) β‰ˆ 𝑦 ↔ 𝑦 = π‘₯))
6765, 15, 66syl2anc 585 . . . . 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 1122 . . 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 3194 . 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 6852 . . 3 (𝑦 = π‘₯ β†’ ((β™―β€˜π‘¦) = 𝑁 ↔ (β™―β€˜π‘₯) = 𝑁))
7271rmo4 3689 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒ*wrmo 3351  {crab 3406  Vcvv 3444   βˆͺ cun 3909   βŠ† wss 3911   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358   β‰ˆ cen 8883   β‰Ό cdom 8884  Fincfn 8886   ≀ cle 11195  β„•cn 12158  β„•0cn0 12418  β™―chash 14236   βˆ₯ cdvds 16141  Basecbs 17088   β†Ύs cress 17117  SubGrpcsubg 18927  odcod 19311  mulGrpcmgp 19901  Unitcui 20073  IDomncidom 20767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-disj 5072  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-ofr 7619  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-tpos 8158  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-omul 8418  df-er 8651  df-ec 8653  df-qs 8657  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-sup 9383  df-inf 9384  df-oi 9451  df-dju 9842  df-card 9880  df-acn 9883  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-xnn0 12491  df-z 12505  df-dec 12624  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-fl 13703  df-mod 13781  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577  df-dvds 16142  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-hom 17162  df-cco 17163  df-0g 17328  df-gsum 17329  df-prds 17334  df-pws 17336  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-submnd 18607  df-grp 18756  df-minusg 18757  df-sbg 18758  df-mulg 18878  df-subg 18930  df-eqg 18932  df-ghm 19011  df-cntz 19102  df-od 19315  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-srg 19923  df-ring 19971  df-cring 19972  df-oppr 20054  df-dvdsr 20075  df-unit 20076  df-invr 20106  df-rnghom 20153  df-subrg 20234  df-lmod 20338  df-lss 20408  df-lsp 20448  df-nzr 20744  df-rlreg 20769  df-domn 20770  df-idom 20771  df-cnfld 20813  df-assa 21275  df-asp 21276  df-ascl 21277  df-psr 21327  df-mvr 21328  df-mpl 21329  df-opsr 21331  df-evls 21498  df-evl 21499  df-psr1 21567  df-vr1 21568  df-ply1 21569  df-coe1 21570  df-evl1 21698  df-mdeg 25433  df-deg1 25434  df-mon1 25511  df-uc1p 25512  df-q1p 25513  df-r1p 25514
This theorem is referenced by:  proot1mul  41569
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