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Theorem elqaalem3 25825
Description: Lemma for elqaa 25826. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
Hypotheses
Ref Expression
elqaa.1 (πœ‘ β†’ 𝐴 ∈ β„‚)
elqaa.2 (πœ‘ β†’ 𝐹 ∈ ((Polyβ€˜β„š) βˆ– {0𝑝}))
elqaa.3 (πœ‘ β†’ (πΉβ€˜π΄) = 0)
elqaa.4 𝐡 = (coeffβ€˜πΉ)
elqaa.5 𝑁 = (π‘˜ ∈ β„•0 ↦ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€}, ℝ, < ))
elqaa.6 𝑅 = (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ))
Assertion
Ref Expression
elqaalem3 (πœ‘ β†’ 𝐴 ∈ 𝔸)
Distinct variable groups:   π‘˜,𝑛,𝐴   𝐡,π‘˜,𝑛   πœ‘,π‘˜   π‘˜,𝑁,𝑛   𝑅,π‘˜
Allowed substitution hints:   πœ‘(𝑛)   𝑅(𝑛)   𝐹(π‘˜,𝑛)
Proof of Theorem elqaalem3
Dummy variables 𝑓 π‘š π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elqaa.1 . 2 (πœ‘ β†’ 𝐴 ∈ β„‚)
2 cnex 11187 . . . . . . . 8 β„‚ ∈ V
32a1i 11 . . . . . . 7 (πœ‘ β†’ β„‚ ∈ V)
4 elqaa.6 . . . . . . . . 9 𝑅 = (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ))
54fvexi 6902 . . . . . . . 8 𝑅 ∈ V
65a1i 11 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 𝑅 ∈ V)
7 fvexd 6903 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (πΉβ€˜π‘§) ∈ V)
8 fconstmpt 5736 . . . . . . . 8 (β„‚ Γ— {𝑅}) = (𝑧 ∈ β„‚ ↦ 𝑅)
98a1i 11 . . . . . . 7 (πœ‘ β†’ (β„‚ Γ— {𝑅}) = (𝑧 ∈ β„‚ ↦ 𝑅))
10 elqaa.2 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 ∈ ((Polyβ€˜β„š) βˆ– {0𝑝}))
1110eldifad 3959 . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„š))
12 plyf 25703 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜β„š) β†’ 𝐹:β„‚βŸΆβ„‚)
1311, 12syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐹:β„‚βŸΆβ„‚)
1413feqmptd 6957 . . . . . . 7 (πœ‘ β†’ 𝐹 = (𝑧 ∈ β„‚ ↦ (πΉβ€˜π‘§)))
153, 6, 7, 9, 14offval2 7686 . . . . . 6 (πœ‘ β†’ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) = (𝑧 ∈ β„‚ ↦ (𝑅 Β· (πΉβ€˜π‘§))))
16 fzfid 13934 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (0...(degβ€˜πΉ)) ∈ Fin)
17 nn0uz 12860 . . . . . . . . . . . . . 14 β„•0 = (β„€β‰₯β€˜0)
18 0zd 12566 . . . . . . . . . . . . . 14 (πœ‘ β†’ 0 ∈ β„€)
19 ssrab2 4076 . . . . . . . . . . . . . . 15 {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} βŠ† β„•
20 fveq2 6888 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ = π‘š β†’ (π΅β€˜π‘˜) = (π΅β€˜π‘š))
2120oveq1d 7420 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = π‘š β†’ ((π΅β€˜π‘˜) Β· 𝑛) = ((π΅β€˜π‘š) Β· 𝑛))
2221eleq1d 2818 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = π‘š β†’ (((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€ ↔ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€))
2322rabbidv 3440 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = π‘š β†’ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€} = {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€})
2423infeq1d 9468 . . . . . . . . . . . . . . . . . 18 (π‘˜ = π‘š β†’ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€}, ℝ, < ) = inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€}, ℝ, < ))
25 elqaa.5 . . . . . . . . . . . . . . . . . 18 𝑁 = (π‘˜ ∈ β„•0 ↦ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘˜) Β· 𝑛) ∈ β„€}, ℝ, < ))
26 ltso 11290 . . . . . . . . . . . . . . . . . . 19 < Or ℝ
2726infex 9484 . . . . . . . . . . . . . . . . . 18 inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€}, ℝ, < ) ∈ V
2824, 25, 27fvmpt 6995 . . . . . . . . . . . . . . . . 17 (π‘š ∈ β„•0 β†’ (π‘β€˜π‘š) = inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€}, ℝ, < ))
2928adantl 482 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) = inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€}, ℝ, < ))
30 nnuz 12861 . . . . . . . . . . . . . . . . . 18 β„• = (β„€β‰₯β€˜1)
3119, 30sseqtri 4017 . . . . . . . . . . . . . . . . 17 {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} βŠ† (β„€β‰₯β€˜1)
32 0z 12565 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ β„€
33 zq 12934 . . . . . . . . . . . . . . . . . . . . . 22 (0 ∈ β„€ β†’ 0 ∈ β„š)
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ β„š
35 elqaa.4 . . . . . . . . . . . . . . . . . . . . . 22 𝐡 = (coeffβ€˜πΉ)
3635coef2 25736 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (Polyβ€˜β„š) ∧ 0 ∈ β„š) β†’ 𝐡:β„•0βŸΆβ„š)
3711, 34, 36sylancl 586 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ 𝐡:β„•0βŸΆβ„š)
3837ffvelcdmda 7083 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π΅β€˜π‘š) ∈ β„š)
39 qmulz 12931 . . . . . . . . . . . . . . . . . . 19 ((π΅β€˜π‘š) ∈ β„š β†’ βˆƒπ‘› ∈ β„• ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€)
4038, 39syl 17 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ βˆƒπ‘› ∈ β„• ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€)
41 rabn0 4384 . . . . . . . . . . . . . . . . . 18 ({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} β‰  βˆ… ↔ βˆƒπ‘› ∈ β„• ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€)
4240, 41sylibr 233 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} β‰  βˆ…)
43 infssuzcl 12912 . . . . . . . . . . . . . . . . 17 (({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} βŠ† (β„€β‰₯β€˜1) ∧ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} β‰  βˆ…) β†’ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€}, ℝ, < ) ∈ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€})
4431, 42, 43sylancr 587 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€}, ℝ, < ) ∈ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€})
4529, 44eqeltrd 2833 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) ∈ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€})
4619, 45sselid 3979 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) ∈ β„•)
47 nnmulcl 12232 . . . . . . . . . . . . . . 15 ((π‘š ∈ β„• ∧ π‘˜ ∈ β„•) β†’ (π‘š Β· π‘˜) ∈ β„•)
4847adantl 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘š ∈ β„• ∧ π‘˜ ∈ β„•)) β†’ (π‘š Β· π‘˜) ∈ β„•)
4917, 18, 46, 48seqf 13985 . . . . . . . . . . . . 13 (πœ‘ β†’ seq0( Β· , 𝑁):β„•0βŸΆβ„•)
50 dgrcl 25738 . . . . . . . . . . . . . 14 (𝐹 ∈ (Polyβ€˜β„š) β†’ (degβ€˜πΉ) ∈ β„•0)
5111, 50syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ (degβ€˜πΉ) ∈ β„•0)
5249, 51ffvelcdmd 7084 . . . . . . . . . . . 12 (πœ‘ β†’ (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ)) ∈ β„•)
534, 52eqeltrid 2837 . . . . . . . . . . 11 (πœ‘ β†’ 𝑅 ∈ β„•)
5453nncnd 12224 . . . . . . . . . 10 (πœ‘ β†’ 𝑅 ∈ β„‚)
5554adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 𝑅 ∈ β„‚)
56 elfznn0 13590 . . . . . . . . . 10 (π‘š ∈ (0...(degβ€˜πΉ)) β†’ π‘š ∈ β„•0)
5735coef3 25737 . . . . . . . . . . . . . 14 (𝐹 ∈ (Polyβ€˜β„š) β†’ 𝐡:β„•0βŸΆβ„‚)
5811, 57syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐡:β„•0βŸΆβ„‚)
5958adantr 481 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 𝐡:β„•0βŸΆβ„‚)
6059ffvelcdmda 7083 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ β„•0) β†’ (π΅β€˜π‘š) ∈ β„‚)
61 expcl 14041 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ π‘š ∈ β„•0) β†’ (π‘§β†‘π‘š) ∈ β„‚)
6261adantll 712 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ β„•0) β†’ (π‘§β†‘π‘š) ∈ β„‚)
6360, 62mulcld 11230 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ β„•0) β†’ ((π΅β€˜π‘š) Β· (π‘§β†‘π‘š)) ∈ β„‚)
6456, 63sylan2 593 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ ((π΅β€˜π‘š) Β· (π‘§β†‘π‘š)) ∈ β„‚)
6516, 55, 64fsummulc2 15726 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (𝑅 Β· Ξ£π‘š ∈ (0...(degβ€˜πΉ))((π΅β€˜π‘š) Β· (π‘§β†‘π‘š))) = Ξ£π‘š ∈ (0...(degβ€˜πΉ))(𝑅 Β· ((π΅β€˜π‘š) Β· (π‘§β†‘π‘š))))
66 eqid 2732 . . . . . . . . . . 11 (degβ€˜πΉ) = (degβ€˜πΉ)
6735, 66coeid2 25744 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜β„š) ∧ 𝑧 ∈ β„‚) β†’ (πΉβ€˜π‘§) = Ξ£π‘š ∈ (0...(degβ€˜πΉ))((π΅β€˜π‘š) Β· (π‘§β†‘π‘š)))
6811, 67sylan 580 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (πΉβ€˜π‘§) = Ξ£π‘š ∈ (0...(degβ€˜πΉ))((π΅β€˜π‘š) Β· (π‘§β†‘π‘š)))
6968oveq2d 7421 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (𝑅 Β· (πΉβ€˜π‘§)) = (𝑅 Β· Ξ£π‘š ∈ (0...(degβ€˜πΉ))((π΅β€˜π‘š) Β· (π‘§β†‘π‘š))))
7055adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ β„•0) β†’ 𝑅 ∈ β„‚)
7170, 60, 62mulassd 11233 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ β„•0) β†’ ((𝑅 Β· (π΅β€˜π‘š)) Β· (π‘§β†‘π‘š)) = (𝑅 Β· ((π΅β€˜π‘š) Β· (π‘§β†‘π‘š))))
7256, 71sylan2 593 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ ((𝑅 Β· (π΅β€˜π‘š)) Β· (π‘§β†‘π‘š)) = (𝑅 Β· ((π΅β€˜π‘š) Β· (π‘§β†‘π‘š))))
7372sumeq2dv 15645 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ Ξ£π‘š ∈ (0...(degβ€˜πΉ))((𝑅 Β· (π΅β€˜π‘š)) Β· (π‘§β†‘π‘š)) = Ξ£π‘š ∈ (0...(degβ€˜πΉ))(𝑅 Β· ((π΅β€˜π‘š) Β· (π‘§β†‘π‘š))))
7465, 69, 733eqtr4d 2782 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (𝑅 Β· (πΉβ€˜π‘§)) = Ξ£π‘š ∈ (0...(degβ€˜πΉ))((𝑅 Β· (π΅β€˜π‘š)) Β· (π‘§β†‘π‘š)))
7574mpteq2dva 5247 . . . . . 6 (πœ‘ β†’ (𝑧 ∈ β„‚ ↦ (𝑅 Β· (πΉβ€˜π‘§))) = (𝑧 ∈ β„‚ ↦ Ξ£π‘š ∈ (0...(degβ€˜πΉ))((𝑅 Β· (π΅β€˜π‘š)) Β· (π‘§β†‘π‘š))))
7615, 75eqtrd 2772 . . . . 5 (πœ‘ β†’ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) = (𝑧 ∈ β„‚ ↦ Ξ£π‘š ∈ (0...(degβ€˜πΉ))((𝑅 Β· (π΅β€˜π‘š)) Β· (π‘§β†‘π‘š))))
77 zsscn 12562 . . . . . . 7 β„€ βŠ† β„‚
7877a1i 11 . . . . . 6 (πœ‘ β†’ β„€ βŠ† β„‚)
7954adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ 𝑅 ∈ β„‚)
8046nncnd 12224 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) ∈ β„‚)
8146nnne0d 12258 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) β‰  0)
8279, 80, 81divcan2d 11988 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ ((π‘β€˜π‘š) Β· (𝑅 / (π‘β€˜π‘š))) = 𝑅)
8382oveq2d 7421 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ ((π΅β€˜π‘š) Β· ((π‘β€˜π‘š) Β· (𝑅 / (π‘β€˜π‘š)))) = ((π΅β€˜π‘š) Β· 𝑅))
8458ffvelcdmda 7083 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π΅β€˜π‘š) ∈ β„‚)
8579, 80, 81divcld 11986 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (𝑅 / (π‘β€˜π‘š)) ∈ β„‚)
8684, 80, 85mulassd 11233 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) Β· (𝑅 / (π‘β€˜π‘š))) = ((π΅β€˜π‘š) Β· ((π‘β€˜π‘š) Β· (𝑅 / (π‘β€˜π‘š)))))
8779, 84mulcomd 11231 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (𝑅 Β· (π΅β€˜π‘š)) = ((π΅β€˜π‘š) Β· 𝑅))
8883, 86, 873eqtr4rd 2783 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (𝑅 Β· (π΅β€˜π‘š)) = (((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) Β· (𝑅 / (π‘β€˜π‘š))))
8956, 88sylan2 593 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ (𝑅 Β· (π΅β€˜π‘š)) = (((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) Β· (𝑅 / (π‘β€˜π‘š))))
90 oveq2 7413 . . . . . . . . . . . . 13 (𝑛 = (π‘β€˜π‘š) β†’ ((π΅β€˜π‘š) Β· 𝑛) = ((π΅β€˜π‘š) Β· (π‘β€˜π‘š)))
9190eleq1d 2818 . . . . . . . . . . . 12 (𝑛 = (π‘β€˜π‘š) β†’ (((π΅β€˜π‘š) Β· 𝑛) ∈ β„€ ↔ ((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) ∈ β„€))
9291elrab 3682 . . . . . . . . . . 11 ((π‘β€˜π‘š) ∈ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} ↔ ((π‘β€˜π‘š) ∈ β„• ∧ ((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) ∈ β„€))
9392simprbi 497 . . . . . . . . . 10 ((π‘β€˜π‘š) ∈ {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} β†’ ((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) ∈ β„€)
9445, 93syl 17 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ ((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) ∈ β„€)
9556, 94sylan2 593 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ ((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) ∈ β„€)
96 elqaa.3 . . . . . . . . . 10 (πœ‘ β†’ (πΉβ€˜π΄) = 0)
97 eqid 2732 . . . . . . . . . 10 (π‘₯ ∈ V, 𝑦 ∈ V ↦ ((π‘₯ Β· 𝑦) mod (π‘β€˜π‘š))) = (π‘₯ ∈ V, 𝑦 ∈ V ↦ ((π‘₯ Β· 𝑦) mod (π‘β€˜π‘š)))
981, 10, 96, 35, 25, 4, 97elqaalem2 25824 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ (𝑅 mod (π‘β€˜π‘š)) = 0)
9953adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ 𝑅 ∈ β„•)
10056, 46sylan2 593 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ (π‘β€˜π‘š) ∈ β„•)
101 nnre 12215 . . . . . . . . . . 11 (𝑅 ∈ β„• β†’ 𝑅 ∈ ℝ)
102 nnrp 12981 . . . . . . . . . . 11 ((π‘β€˜π‘š) ∈ β„• β†’ (π‘β€˜π‘š) ∈ ℝ+)
103 mod0 13837 . . . . . . . . . . 11 ((𝑅 ∈ ℝ ∧ (π‘β€˜π‘š) ∈ ℝ+) β†’ ((𝑅 mod (π‘β€˜π‘š)) = 0 ↔ (𝑅 / (π‘β€˜π‘š)) ∈ β„€))
104101, 102, 103syl2an 596 . . . . . . . . . 10 ((𝑅 ∈ β„• ∧ (π‘β€˜π‘š) ∈ β„•) β†’ ((𝑅 mod (π‘β€˜π‘š)) = 0 ↔ (𝑅 / (π‘β€˜π‘š)) ∈ β„€))
10599, 100, 104syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ ((𝑅 mod (π‘β€˜π‘š)) = 0 ↔ (𝑅 / (π‘β€˜π‘š)) ∈ β„€))
10698, 105mpbid 231 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ (𝑅 / (π‘β€˜π‘š)) ∈ β„€)
10795, 106zmulcld 12668 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ (((π΅β€˜π‘š) Β· (π‘β€˜π‘š)) Β· (𝑅 / (π‘β€˜π‘š))) ∈ β„€)
10889, 107eqeltrd 2833 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (0...(degβ€˜πΉ))) β†’ (𝑅 Β· (π΅β€˜π‘š)) ∈ β„€)
10978, 51, 108elplyd 25707 . . . . 5 (πœ‘ β†’ (𝑧 ∈ β„‚ ↦ Ξ£π‘š ∈ (0...(degβ€˜πΉ))((𝑅 Β· (π΅β€˜π‘š)) Β· (π‘§β†‘π‘š))) ∈ (Polyβ€˜β„€))
11076, 109eqeltrd 2833 . . . 4 (πœ‘ β†’ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∈ (Polyβ€˜β„€))
111 eldifsn 4789 . . . . . . 7 (𝐹 ∈ ((Polyβ€˜β„š) βˆ– {0𝑝}) ↔ (𝐹 ∈ (Polyβ€˜β„š) ∧ 𝐹 β‰  0𝑝))
11210, 111sylib 217 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ (Polyβ€˜β„š) ∧ 𝐹 β‰  0𝑝))
113112simprd 496 . . . . 5 (πœ‘ β†’ 𝐹 β‰  0𝑝)
114 oveq1 7412 . . . . . . 7 (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) = 0𝑝 β†’ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∘f / (β„‚ Γ— {𝑅})) = (0𝑝 ∘f / (β„‚ Γ— {𝑅})))
11513ffvelcdmda 7083 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (πΉβ€˜π‘§) ∈ β„‚)
11653nnne0d 12258 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑅 β‰  0)
117116adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 𝑅 β‰  0)
118115, 55, 117divcan3d 11991 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ ((𝑅 Β· (πΉβ€˜π‘§)) / 𝑅) = (πΉβ€˜π‘§))
119118mpteq2dva 5247 . . . . . . . . 9 (πœ‘ β†’ (𝑧 ∈ β„‚ ↦ ((𝑅 Β· (πΉβ€˜π‘§)) / 𝑅)) = (𝑧 ∈ β„‚ ↦ (πΉβ€˜π‘§)))
120 ovexd 7440 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (𝑅 Β· (πΉβ€˜π‘§)) ∈ V)
1213, 120, 6, 15, 9offval2 7686 . . . . . . . . 9 (πœ‘ β†’ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∘f / (β„‚ Γ— {𝑅})) = (𝑧 ∈ β„‚ ↦ ((𝑅 Β· (πΉβ€˜π‘§)) / 𝑅)))
122119, 121, 143eqtr4d 2782 . . . . . . . 8 (πœ‘ β†’ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∘f / (β„‚ Γ— {𝑅})) = 𝐹)
12354, 116div0d 11985 . . . . . . . . . 10 (πœ‘ β†’ (0 / 𝑅) = 0)
124123mpteq2dv 5249 . . . . . . . . 9 (πœ‘ β†’ (𝑧 ∈ β„‚ ↦ (0 / 𝑅)) = (𝑧 ∈ β„‚ ↦ 0))
125 0cnd 11203 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 0 ∈ β„‚)
126 df-0p 25178 . . . . . . . . . . . 12 0𝑝 = (β„‚ Γ— {0})
127 fconstmpt 5736 . . . . . . . . . . . 12 (β„‚ Γ— {0}) = (𝑧 ∈ β„‚ ↦ 0)
128126, 127eqtri 2760 . . . . . . . . . . 11 0𝑝 = (𝑧 ∈ β„‚ ↦ 0)
129128a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 0𝑝 = (𝑧 ∈ β„‚ ↦ 0))
1303, 125, 6, 129, 9offval2 7686 . . . . . . . . 9 (πœ‘ β†’ (0𝑝 ∘f / (β„‚ Γ— {𝑅})) = (𝑧 ∈ β„‚ ↦ (0 / 𝑅)))
131124, 130, 1293eqtr4d 2782 . . . . . . . 8 (πœ‘ β†’ (0𝑝 ∘f / (β„‚ Γ— {𝑅})) = 0𝑝)
132122, 131eqeq12d 2748 . . . . . . 7 (πœ‘ β†’ ((((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∘f / (β„‚ Γ— {𝑅})) = (0𝑝 ∘f / (β„‚ Γ— {𝑅})) ↔ 𝐹 = 0𝑝))
133114, 132imbitrid 243 . . . . . 6 (πœ‘ β†’ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) = 0𝑝 β†’ 𝐹 = 0𝑝))
134133necon3d 2961 . . . . 5 (πœ‘ β†’ (𝐹 β‰  0𝑝 β†’ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) β‰  0𝑝))
135113, 134mpd 15 . . . 4 (πœ‘ β†’ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) β‰  0𝑝)
136 eldifsn 4789 . . . 4 (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∈ ((Polyβ€˜β„€) βˆ– {0𝑝}) ↔ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∈ (Polyβ€˜β„€) ∧ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) β‰  0𝑝))
137110, 135, 136sylanbrc 583 . . 3 (πœ‘ β†’ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∈ ((Polyβ€˜β„€) βˆ– {0𝑝}))
1385fconst 6774 . . . . . . 7 (β„‚ Γ— {𝑅}):β„‚βŸΆ{𝑅}
139 ffn 6714 . . . . . . 7 ((β„‚ Γ— {𝑅}):β„‚βŸΆ{𝑅} β†’ (β„‚ Γ— {𝑅}) Fn β„‚)
140138, 139mp1i 13 . . . . . 6 (πœ‘ β†’ (β„‚ Γ— {𝑅}) Fn β„‚)
14113ffnd 6715 . . . . . 6 (πœ‘ β†’ 𝐹 Fn β„‚)
142 inidm 4217 . . . . . 6 (β„‚ ∩ β„‚) = β„‚
1435fvconst2 7201 . . . . . . 7 (𝐴 ∈ β„‚ β†’ ((β„‚ Γ— {𝑅})β€˜π΄) = 𝑅)
144143adantl 482 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ β„‚) β†’ ((β„‚ Γ— {𝑅})β€˜π΄) = 𝑅)
14596adantr 481 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ β„‚) β†’ (πΉβ€˜π΄) = 0)
146140, 141, 3, 3, 142, 144, 145ofval 7677 . . . . 5 ((πœ‘ ∧ 𝐴 ∈ β„‚) β†’ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹)β€˜π΄) = (𝑅 Β· 0))
1471, 146mpdan 685 . . . 4 (πœ‘ β†’ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹)β€˜π΄) = (𝑅 Β· 0))
14854mul01d 11409 . . . 4 (πœ‘ β†’ (𝑅 Β· 0) = 0)
149147, 148eqtrd 2772 . . 3 (πœ‘ β†’ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹)β€˜π΄) = 0)
150 fveq1 6887 . . . . 5 (𝑓 = ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) β†’ (π‘“β€˜π΄) = (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹)β€˜π΄))
151150eqeq1d 2734 . . . 4 (𝑓 = ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) β†’ ((π‘“β€˜π΄) = 0 ↔ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹)β€˜π΄) = 0))
152151rspcev 3612 . . 3 ((((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) ∈ ((Polyβ€˜β„€) βˆ– {0𝑝}) ∧ (((β„‚ Γ— {𝑅}) ∘f Β· 𝐹)β€˜π΄) = 0) β†’ βˆƒπ‘“ ∈ ((Polyβ€˜β„€) βˆ– {0𝑝})(π‘“β€˜π΄) = 0)
153137, 149, 152syl2anc 584 . 2 (πœ‘ β†’ βˆƒπ‘“ ∈ ((Polyβ€˜β„€) βˆ– {0𝑝})(π‘“β€˜π΄) = 0)
154 elaa 25820 . 2 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ β„‚ ∧ βˆƒπ‘“ ∈ ((Polyβ€˜β„€) βˆ– {0𝑝})(π‘“β€˜π΄) = 0))
1551, 153, 154sylanbrc 583 1 (πœ‘ β†’ 𝐴 ∈ 𝔸)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  {csn 4627   ↦ cmpt 5230   Γ— cxp 5673   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ∘f cof 7664  infcinf 9432  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   Β· cmul 11111   < clt 11244   / cdiv 11867  β„•cn 12208  β„•0cn0 12468  β„€cz 12554  β„€β‰₯cuz 12818  β„šcq 12928  β„+crp 12970  ...cfz 13480   mod cmo 13830  seqcseq 13962  β†‘cexp 14023  Ξ£csu 15628  0𝑝c0p 25177  Polycply 25689  coeffccoe 25691  degcdgr 25692  π”Έcaa 25818
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-0p 25178  df-ply 25693  df-coe 25695  df-dgr 25696  df-aa 25819
This theorem is referenced by:  elqaa  25826
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