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Theorem idomrootle 26062
Description: No element of an integral domain can have more than 𝑁 𝑁-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)
Hypotheses
Ref Expression
idomrootle.b 𝐡 = (Baseβ€˜π‘…)
idomrootle.e ↑ = (.gβ€˜(mulGrpβ€˜π‘…))
Assertion
Ref Expression
idomrootle ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜{𝑦 ∈ 𝐡 ∣ (𝑁 ↑ 𝑦) = 𝑋}) ≀ 𝑁)
Distinct variable groups:   𝑦,𝐡   𝑦,𝑁   𝑦,𝑅   𝑦,𝑋
Allowed substitution hint:   ↑ (𝑦)

Proof of Theorem idomrootle
StepHypRef Expression
1 eqid 2726 . . 3 (Poly1β€˜π‘…) = (Poly1β€˜π‘…)
2 eqid 2726 . . 3 (Baseβ€˜(Poly1β€˜π‘…)) = (Baseβ€˜(Poly1β€˜π‘…))
3 eqid 2726 . . 3 ( deg1 β€˜π‘…) = ( deg1 β€˜π‘…)
4 eqid 2726 . . 3 (eval1β€˜π‘…) = (eval1β€˜π‘…)
5 eqid 2726 . . 3 (0gβ€˜π‘…) = (0gβ€˜π‘…)
6 eqid 2726 . . 3 (0gβ€˜(Poly1β€˜π‘…)) = (0gβ€˜(Poly1β€˜π‘…))
7 simp1 1133 . . 3 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑅 ∈ IDomn)
8 isidom 21216 . . . . . . . . 9 (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
98simplbi 497 . . . . . . . 8 (𝑅 ∈ IDomn β†’ 𝑅 ∈ CRing)
107, 9syl 17 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑅 ∈ CRing)
11 crngring 20150 . . . . . . 7 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
1210, 11syl 17 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑅 ∈ Ring)
131ply1ring 22121 . . . . . 6 (𝑅 ∈ Ring β†’ (Poly1β€˜π‘…) ∈ Ring)
1412, 13syl 17 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (Poly1β€˜π‘…) ∈ Ring)
15 ringgrp 20143 . . . . 5 ((Poly1β€˜π‘…) ∈ Ring β†’ (Poly1β€˜π‘…) ∈ Grp)
1614, 15syl 17 . . . 4 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (Poly1β€˜π‘…) ∈ Grp)
17 eqid 2726 . . . . . . . 8 (mulGrpβ€˜(Poly1β€˜π‘…)) = (mulGrpβ€˜(Poly1β€˜π‘…))
1817ringmgp 20144 . . . . . . 7 ((Poly1β€˜π‘…) ∈ Ring β†’ (mulGrpβ€˜(Poly1β€˜π‘…)) ∈ Mnd)
1914, 18syl 17 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (mulGrpβ€˜(Poly1β€˜π‘…)) ∈ Mnd)
20 mndmgm 18674 . . . . . 6 ((mulGrpβ€˜(Poly1β€˜π‘…)) ∈ Mnd β†’ (mulGrpβ€˜(Poly1β€˜π‘…)) ∈ Mgm)
2119, 20syl 17 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (mulGrpβ€˜(Poly1β€˜π‘…)) ∈ Mgm)
22 simp3 1135 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ β„•)
23 eqid 2726 . . . . . . 7 (var1β€˜π‘…) = (var1β€˜π‘…)
2423, 1, 2vr1cl 22091 . . . . . 6 (𝑅 ∈ Ring β†’ (var1β€˜π‘…) ∈ (Baseβ€˜(Poly1β€˜π‘…)))
2512, 24syl 17 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (var1β€˜π‘…) ∈ (Baseβ€˜(Poly1β€˜π‘…)))
2617, 2mgpbas 20045 . . . . . 6 (Baseβ€˜(Poly1β€˜π‘…)) = (Baseβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))
27 eqid 2726 . . . . . 6 (.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…))) = (.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))
2826, 27mulgnncl 19016 . . . . 5 (((mulGrpβ€˜(Poly1β€˜π‘…)) ∈ Mgm ∧ 𝑁 ∈ β„• ∧ (var1β€˜π‘…) ∈ (Baseβ€˜(Poly1β€˜π‘…))) β†’ (𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…)) ∈ (Baseβ€˜(Poly1β€˜π‘…)))
2921, 22, 25, 28syl3anc 1368 . . . 4 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…)) ∈ (Baseβ€˜(Poly1β€˜π‘…)))
30 eqid 2726 . . . . . . 7 (algScβ€˜(Poly1β€˜π‘…)) = (algScβ€˜(Poly1β€˜π‘…))
31 idomrootle.b . . . . . . 7 𝐡 = (Baseβ€˜π‘…)
321, 30, 31, 2ply1sclf 22159 . . . . . 6 (𝑅 ∈ Ring β†’ (algScβ€˜(Poly1β€˜π‘…)):𝐡⟢(Baseβ€˜(Poly1β€˜π‘…)))
3312, 32syl 17 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (algScβ€˜(Poly1β€˜π‘…)):𝐡⟢(Baseβ€˜(Poly1β€˜π‘…)))
34 simp2 1134 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑋 ∈ 𝐡)
3533, 34ffvelcdmd 7081 . . . 4 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ ((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹) ∈ (Baseβ€˜(Poly1β€˜π‘…)))
36 eqid 2726 . . . . 5 (-gβ€˜(Poly1β€˜π‘…)) = (-gβ€˜(Poly1β€˜π‘…))
372, 36grpsubcl 18948 . . . 4 (((Poly1β€˜π‘…) ∈ Grp ∧ (𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…)) ∈ (Baseβ€˜(Poly1β€˜π‘…)) ∧ ((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹) ∈ (Baseβ€˜(Poly1β€˜π‘…))) β†’ ((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ∈ (Baseβ€˜(Poly1β€˜π‘…)))
3816, 29, 35, 37syl3anc 1368 . . 3 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ ((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ∈ (Baseβ€˜(Poly1β€˜π‘…)))
393, 1, 2deg1xrcl 25973 . . . . . . . . . 10 (((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹) ∈ (Baseβ€˜(Poly1β€˜π‘…)) β†’ (( deg1 β€˜π‘…)β€˜((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ∈ ℝ*)
4035, 39syl 17 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ∈ ℝ*)
41 0xr 11265 . . . . . . . . . 10 0 ∈ ℝ*
4241a1i 11 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 0 ∈ ℝ*)
43 nnre 12223 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ 𝑁 ∈ ℝ)
4443rexrd 11268 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ 𝑁 ∈ ℝ*)
45443ad2ant3 1132 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ ℝ*)
463, 1, 31, 30deg1sclle 26003 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (( deg1 β€˜π‘…)β€˜((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ≀ 0)
4712, 34, 46syl2anc 583 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ≀ 0)
48 nngt0 12247 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ 0 < 𝑁)
49483ad2ant3 1132 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 0 < 𝑁)
5040, 42, 45, 47, 49xrlelttrd 13145 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) < 𝑁)
518simprbi 496 . . . . . . . . . . 11 (𝑅 ∈ IDomn β†’ 𝑅 ∈ Domn)
52 domnnzr 21205 . . . . . . . . . . 11 (𝑅 ∈ Domn β†’ 𝑅 ∈ NzRing)
5351, 52syl 17 . . . . . . . . . 10 (𝑅 ∈ IDomn β†’ 𝑅 ∈ NzRing)
547, 53syl 17 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑅 ∈ NzRing)
55 nnnn0 12483 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
56553ad2ant3 1132 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ β„•0)
573, 1, 23, 17, 27deg1pw 26011 . . . . . . . . 9 ((𝑅 ∈ NzRing ∧ 𝑁 ∈ β„•0) β†’ (( deg1 β€˜π‘…)β€˜(𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))) = 𝑁)
5854, 56, 57syl2anc 583 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜(𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))) = 𝑁)
5950, 58breqtrrd 5169 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) < (( deg1 β€˜π‘…)β€˜(𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))))
601, 3, 12, 2, 36, 29, 35, 59deg1sub 25999 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) = (( deg1 β€˜π‘…)β€˜(𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))))
6160, 58eqtrd 2766 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) = 𝑁)
6261, 56eqeltrd 2827 . . . 4 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (( deg1 β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) ∈ β„•0)
633, 1, 6, 2deg1nn0clb 25981 . . . . 5 ((𝑅 ∈ Ring ∧ ((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ∈ (Baseβ€˜(Poly1β€˜π‘…))) β†’ (((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) β‰  (0gβ€˜(Poly1β€˜π‘…)) ↔ (( deg1 β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) ∈ β„•0))
6412, 38, 63syl2anc 583 . . . 4 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) β‰  (0gβ€˜(Poly1β€˜π‘…)) ↔ (( deg1 β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) ∈ β„•0))
6562, 64mpbird 257 . . 3 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ ((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) β‰  (0gβ€˜(Poly1β€˜π‘…)))
661, 2, 3, 4, 5, 6, 7, 38, 65fta1g 26059 . 2 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜(β—‘((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) β€œ {(0gβ€˜π‘…)})) ≀ (( deg1 β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))))
67 eqid 2726 . . . . . . 7 (𝑅 ↑s 𝐡) = (𝑅 ↑s 𝐡)
68 eqid 2726 . . . . . . 7 (Baseβ€˜(𝑅 ↑s 𝐡)) = (Baseβ€˜(𝑅 ↑s 𝐡))
6931fvexi 6899 . . . . . . . 8 𝐡 ∈ V
7069a1i 11 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝐡 ∈ V)
714, 1, 67, 31evl1rhm 22206 . . . . . . . . . 10 (𝑅 ∈ CRing β†’ (eval1β€˜π‘…) ∈ ((Poly1β€˜π‘…) RingHom (𝑅 ↑s 𝐡)))
7210, 71syl 17 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (eval1β€˜π‘…) ∈ ((Poly1β€˜π‘…) RingHom (𝑅 ↑s 𝐡)))
732, 68rhmf 20387 . . . . . . . . 9 ((eval1β€˜π‘…) ∈ ((Poly1β€˜π‘…) RingHom (𝑅 ↑s 𝐡)) β†’ (eval1β€˜π‘…):(Baseβ€˜(Poly1β€˜π‘…))⟢(Baseβ€˜(𝑅 ↑s 𝐡)))
7472, 73syl 17 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (eval1β€˜π‘…):(Baseβ€˜(Poly1β€˜π‘…))⟢(Baseβ€˜(𝑅 ↑s 𝐡)))
7574, 38ffvelcdmd 7081 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ ((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) ∈ (Baseβ€˜(𝑅 ↑s 𝐡)))
7667, 31, 68, 7, 70, 75pwselbas 17444 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ ((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))):𝐡⟢𝐡)
7776ffnd 6712 . . . . 5 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ ((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) Fn 𝐡)
78 fniniseg2 7057 . . . . 5 (((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) Fn 𝐡 β†’ (β—‘((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) β€œ {(0gβ€˜π‘…)}) = {𝑦 ∈ 𝐡 ∣ (((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)))β€˜π‘¦) = (0gβ€˜π‘…)})
7977, 78syl 17 . . . 4 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (β—‘((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) β€œ {(0gβ€˜π‘…)}) = {𝑦 ∈ 𝐡 ∣ (((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)))β€˜π‘¦) = (0gβ€˜π‘…)})
8010adantr 480 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ 𝑅 ∈ CRing)
81 simpr 484 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 ∈ 𝐡)
824, 23, 31, 1, 2, 80, 81evl1vard 22211 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ ((var1β€˜π‘…) ∈ (Baseβ€˜(Poly1β€˜π‘…)) ∧ (((eval1β€˜π‘…)β€˜(var1β€˜π‘…))β€˜π‘¦) = 𝑦))
83 idomrootle.e . . . . . . . . . 10 ↑ = (.gβ€˜(mulGrpβ€˜π‘…))
84 simpl3 1190 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ 𝑁 ∈ β„•)
8584, 55syl 17 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ 𝑁 ∈ β„•0)
864, 1, 31, 2, 80, 81, 82, 27, 83, 85evl1expd 22219 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ ((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…)) ∈ (Baseβ€˜(Poly1β€˜π‘…)) ∧ (((eval1β€˜π‘…)β€˜(𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…)))β€˜π‘¦) = (𝑁 ↑ 𝑦)))
87 simpl2 1189 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
884, 1, 31, 30, 2, 80, 87, 81evl1scad 22209 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ (((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹) ∈ (Baseβ€˜(Poly1β€˜π‘…)) ∧ (((eval1β€˜π‘…)β€˜((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))β€˜π‘¦) = 𝑋))
89 eqid 2726 . . . . . . . . 9 (-gβ€˜π‘…) = (-gβ€˜π‘…)
904, 1, 31, 2, 80, 81, 86, 88, 36, 89evl1subd 22216 . . . . . . . 8 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ (((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)) ∈ (Baseβ€˜(Poly1β€˜π‘…)) ∧ (((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)))β€˜π‘¦) = ((𝑁 ↑ 𝑦)(-gβ€˜π‘…)𝑋)))
9190simprd 495 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ (((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)))β€˜π‘¦) = ((𝑁 ↑ 𝑦)(-gβ€˜π‘…)𝑋))
9291eqeq1d 2728 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ ((((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)))β€˜π‘¦) = (0gβ€˜π‘…) ↔ ((𝑁 ↑ 𝑦)(-gβ€˜π‘…)𝑋) = (0gβ€˜π‘…)))
93 ringgrp 20143 . . . . . . . . 9 (𝑅 ∈ Ring β†’ 𝑅 ∈ Grp)
9412, 93syl 17 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ 𝑅 ∈ Grp)
9594adantr 480 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ 𝑅 ∈ Grp)
96 eqid 2726 . . . . . . . . . . . 12 (mulGrpβ€˜π‘…) = (mulGrpβ€˜π‘…)
9796ringmgp 20144 . . . . . . . . . . 11 (𝑅 ∈ Ring β†’ (mulGrpβ€˜π‘…) ∈ Mnd)
9812, 97syl 17 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (mulGrpβ€˜π‘…) ∈ Mnd)
9998adantr 480 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ (mulGrpβ€˜π‘…) ∈ Mnd)
100 mndmgm 18674 . . . . . . . . 9 ((mulGrpβ€˜π‘…) ∈ Mnd β†’ (mulGrpβ€˜π‘…) ∈ Mgm)
10199, 100syl 17 . . . . . . . 8 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ (mulGrpβ€˜π‘…) ∈ Mgm)
10296, 31mgpbas 20045 . . . . . . . . 9 𝐡 = (Baseβ€˜(mulGrpβ€˜π‘…))
103102, 83mulgnncl 19016 . . . . . . . 8 (((mulGrpβ€˜π‘…) ∈ Mgm ∧ 𝑁 ∈ β„• ∧ 𝑦 ∈ 𝐡) β†’ (𝑁 ↑ 𝑦) ∈ 𝐡)
104101, 84, 81, 103syl3anc 1368 . . . . . . 7 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ (𝑁 ↑ 𝑦) ∈ 𝐡)
10531, 5, 89grpsubeq0 18954 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝑁 ↑ 𝑦) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (((𝑁 ↑ 𝑦)(-gβ€˜π‘…)𝑋) = (0gβ€˜π‘…) ↔ (𝑁 ↑ 𝑦) = 𝑋))
10695, 104, 87, 105syl3anc 1368 . . . . . 6 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ (((𝑁 ↑ 𝑦)(-gβ€˜π‘…)𝑋) = (0gβ€˜π‘…) ↔ (𝑁 ↑ 𝑦) = 𝑋))
10792, 106bitrd 279 . . . . 5 (((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) ∧ 𝑦 ∈ 𝐡) β†’ ((((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)))β€˜π‘¦) = (0gβ€˜π‘…) ↔ (𝑁 ↑ 𝑦) = 𝑋))
108107rabbidva 3433 . . . 4 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ {𝑦 ∈ 𝐡 ∣ (((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹)))β€˜π‘¦) = (0gβ€˜π‘…)} = {𝑦 ∈ 𝐡 ∣ (𝑁 ↑ 𝑦) = 𝑋})
10979, 108eqtrd 2766 . . 3 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (β—‘((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) β€œ {(0gβ€˜π‘…)}) = {𝑦 ∈ 𝐡 ∣ (𝑁 ↑ 𝑦) = 𝑋})
110109fveq2d 6889 . 2 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜(β—‘((eval1β€˜π‘…)β€˜((𝑁(.gβ€˜(mulGrpβ€˜(Poly1β€˜π‘…)))(var1β€˜π‘…))(-gβ€˜(Poly1β€˜π‘…))((algScβ€˜(Poly1β€˜π‘…))β€˜π‘‹))) β€œ {(0gβ€˜π‘…)})) = (β™―β€˜{𝑦 ∈ 𝐡 ∣ (𝑁 ↑ 𝑦) = 𝑋}))
11166, 110, 613brtr3d 5172 1 ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐡 ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜{𝑦 ∈ 𝐡 ∣ (𝑁 ↑ 𝑦) = 𝑋}) ≀ 𝑁)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426  Vcvv 3468  {csn 4623   class class class wbr 5141  β—‘ccnv 5668   β€œ cima 5672   Fn wfn 6532  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  β„*cxr 11251   < clt 11252   ≀ cle 11253  β„•cn 12216  β„•0cn0 12476  β™―chash 14295  Basecbs 17153  0gc0g 17394   ↑s cpws 17401  Mgmcmgm 18571  Mndcmnd 18667  Grpcgrp 18863  -gcsg 18865  .gcmg 18995  mulGrpcmgp 20039  Ringcrg 20138  CRingccrg 20139   RingHom crh 20371  NzRingcnzr 20414  Domncdomn 21190  IDomncidom 21191  algSccascl 21747  var1cv1 22050  Poly1cpl1 22051  eval1ce1 22188   deg1 cdg1 25942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-ofr 7668  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  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-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-hom 17230  df-cco 17231  df-0g 17396  df-gsum 17397  df-prds 17402  df-pws 17404  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18996  df-subg 19050  df-ghm 19139  df-cntz 19233  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-srg 20092  df-ring 20140  df-cring 20141  df-oppr 20236  df-dvdsr 20259  df-unit 20260  df-invr 20290  df-rhm 20374  df-nzr 20415  df-subrng 20446  df-subrg 20471  df-lmod 20708  df-lss 20779  df-lsp 20819  df-rlreg 21193  df-domn 21194  df-idom 21195  df-cnfld 21241  df-assa 21748  df-asp 21749  df-ascl 21750  df-psr 21803  df-mvr 21804  df-mpl 21805  df-opsr 21807  df-evls 21977  df-evl 21978  df-psr1 22054  df-vr1 22055  df-ply1 22056  df-coe1 22057  df-evl1 22190  df-mdeg 25943  df-deg1 25944  df-mon1 26021  df-uc1p 26022  df-q1p 26023  df-r1p 26024
This theorem is referenced by:  idomodle  42520
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