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

Theorem idomsubgmo 41940
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 6905 . . . . . . . . 9 (Baseβ€˜πΊ) ∈ V
21rabex 5333 . . . . . . . 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 19007 . . . . . . . . . . 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 12532 . . . . . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . 15 𝑦 ∈ V
13 hashclb 14318 . . . . . . . . . . . . . . 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 19439 . . . . . . . . . . . 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 5175 . . . . . . . . . 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 4072 . . . . . . . . 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 19007 . . . . . . . . . . 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 3479 . . . . . . . . . . . . . . 15 π‘₯ ∈ V
31 hashclb 14318 . . . . . . . . . . . . . . 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 19439 . . . . . . . . . . . 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 5175 . . . . . . . . . 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 4072 . . . . . . . . 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 4187 . . . . . . . 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 8996 . . . . . . . 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 41938 . . . . . . . . . 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 5177 . . . . . . . 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 14296 . . . . . . . . . 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 14339 . . . . . . . . 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 9003 . . . . . . 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 7733 . . . . . . 7 (𝑦 βˆͺ π‘₯) ∈ V
57 ssun1 4173 . . . . . . 7 𝑦 βŠ† (𝑦 βˆͺ π‘₯)
58 ssdomg 8996 . . . . . . 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 9093 . . . . . 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 14307 . . . . . . . 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 41939 . . . . . 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 3201 . 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 6901 . . 3 (𝑦 = π‘₯ β†’ ((β™―β€˜π‘¦) = 𝑁 ↔ (β™―β€˜π‘₯) = 𝑁))
7271rmo4 3727 . 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 3062  βˆƒ*wrmo 3376  {crab 3433  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409   β‰ˆ cen 8936   β‰Ό cdom 8937  Fincfn 8939   ≀ cle 11249  β„•cn 12212  β„•0cn0 12472  β™―chash 14290   βˆ₯ cdvds 16197  Basecbs 17144   β†Ύs cress 17173  SubGrpcsubg 19000  odcod 19392  mulGrpcmgp 19987  Unitcui 20169  IDomncidom 20897
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-ofr 7671  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-ec 8705  df-qs 8709  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-sup 9437  df-inf 9438  df-oi 9505  df-dju 9896  df-card 9934  df-acn 9937  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-xnn0 12545  df-z 12559  df-dec 12678  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-dvds 16198  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-0g 17387  df-gsum 17388  df-prds 17393  df-pws 17395  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-eqg 19005  df-ghm 19090  df-cntz 19181  df-od 19396  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-srg 20010  df-ring 20058  df-cring 20059  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-rnghom 20251  df-nzr 20292  df-subrg 20317  df-lmod 20473  df-lss 20543  df-lsp 20583  df-rlreg 20899  df-domn 20900  df-idom 20901  df-cnfld 20945  df-assa 21408  df-asp 21409  df-ascl 21410  df-psr 21462  df-mvr 21463  df-mpl 21464  df-opsr 21466  df-evls 21635  df-evl 21636  df-psr1 21704  df-vr1 21705  df-ply1 21706  df-coe1 21707  df-evl1 21835  df-mdeg 25570  df-deg1 25571  df-mon1 25648  df-uc1p 25649  df-q1p 25650  df-r1p 25651
This theorem is referenced by:  proot1mul  41941
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