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Theorem elqaalem3 25841
Description: Lemma for elqaa 25842. (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 11193 . . . . . . . 8 β„‚ ∈ V
32a1i 11 . . . . . . 7 (πœ‘ β†’ β„‚ ∈ V)
4 elqaa.6 . . . . . . . . 9 𝑅 = (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ))
54fvexi 6905 . . . . . . . 8 𝑅 ∈ V
65a1i 11 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 𝑅 ∈ V)
7 fvexd 6906 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (πΉβ€˜π‘§) ∈ V)
8 fconstmpt 5738 . . . . . . . 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 3960 . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ (Polyβ€˜β„š))
12 plyf 25719 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜β„š) β†’ 𝐹:β„‚βŸΆβ„‚)
1311, 12syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐹:β„‚βŸΆβ„‚)
1413feqmptd 6960 . . . . . . 7 (πœ‘ β†’ 𝐹 = (𝑧 ∈ β„‚ ↦ (πΉβ€˜π‘§)))
153, 6, 7, 9, 14offval2 7692 . . . . . 6 (πœ‘ β†’ ((β„‚ Γ— {𝑅}) ∘f Β· 𝐹) = (𝑧 ∈ β„‚ ↦ (𝑅 Β· (πΉβ€˜π‘§))))
16 fzfid 13940 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (0...(degβ€˜πΉ)) ∈ Fin)
17 nn0uz 12866 . . . . . . . . . . . . . 14 β„•0 = (β„€β‰₯β€˜0)
18 0zd 12572 . . . . . . . . . . . . . 14 (πœ‘ β†’ 0 ∈ β„€)
19 ssrab2 4077 . . . . . . . . . . . . . . 15 {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} βŠ† β„•
20 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ = π‘š β†’ (π΅β€˜π‘˜) = (π΅β€˜π‘š))
2120oveq1d 7426 . . . . . . . . . . . . . . . . . . . . 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 9474 . . . . . . . . . . . . . . . . . 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 11296 . . . . . . . . . . . . . . . . . . 19 < Or ℝ
2726infex 9490 . . . . . . . . . . . . . . . . . 18 inf({𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€}, ℝ, < ) ∈ V
2824, 25, 27fvmpt 6998 . . . . . . . . . . . . . . . . 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 12867 . . . . . . . . . . . . . . . . . 18 β„• = (β„€β‰₯β€˜1)
3119, 30sseqtri 4018 . . . . . . . . . . . . . . . . 17 {𝑛 ∈ β„• ∣ ((π΅β€˜π‘š) Β· 𝑛) ∈ β„€} βŠ† (β„€β‰₯β€˜1)
32 0z 12571 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ β„€
33 zq 12940 . . . . . . . . . . . . . . . . . . . . . 22 (0 ∈ β„€ β†’ 0 ∈ β„š)
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ β„š
35 elqaa.4 . . . . . . . . . . . . . . . . . . . . . 22 𝐡 = (coeffβ€˜πΉ)
3635coef2 25752 . . . . . . . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π΅β€˜π‘š) ∈ β„š)
39 qmulz 12937 . . . . . . . . . . . . . . . . . . 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 4385 . . . . . . . . . . . . . . . . . 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 12918 . . . . . . . . . . . . . . . . 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 3980 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) ∈ β„•)
47 nnmulcl 12238 . . . . . . . . . . . . . . 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 13991 . . . . . . . . . . . . 13 (πœ‘ β†’ seq0( Β· , 𝑁):β„•0βŸΆβ„•)
50 dgrcl 25754 . . . . . . . . . . . . . 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 7087 . . . . . . . . . . . 12 (πœ‘ β†’ (seq0( Β· , 𝑁)β€˜(degβ€˜πΉ)) ∈ β„•)
534, 52eqeltrid 2837 . . . . . . . . . . 11 (πœ‘ β†’ 𝑅 ∈ β„•)
5453nncnd 12230 . . . . . . . . . 10 (πœ‘ β†’ 𝑅 ∈ β„‚)
5554adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 𝑅 ∈ β„‚)
56 elfznn0 13596 . . . . . . . . . 10 (π‘š ∈ (0...(degβ€˜πΉ)) β†’ π‘š ∈ β„•0)
5735coef3 25753 . . . . . . . . . . . . . 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 7086 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ β„‚) ∧ π‘š ∈ β„•0) β†’ (π΅β€˜π‘š) ∈ β„‚)
61 expcl 14047 . . . . . . . . . . . 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 11236 . . . . . . . . . 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 15732 . . . . . . . 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 25760 . . . . . . . . . 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 7427 . . . . . . . 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 11239 . . . . . . . . . 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 15651 . . . . . . . 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 5248 . . . . . 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 12568 . . . . . . 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 12230 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) ∈ β„‚)
8146nnne0d 12264 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π‘β€˜π‘š) β‰  0)
8279, 80, 81divcan2d 11994 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ ((π‘β€˜π‘š) Β· (𝑅 / (π‘β€˜π‘š))) = 𝑅)
8382oveq2d 7427 . . . . . . . . 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 7086 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (π΅β€˜π‘š) ∈ β„‚)
8579, 80, 81divcld 11992 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ β„•0) β†’ (𝑅 / (π‘β€˜π‘š)) ∈ β„‚)
8684, 80, 85mulassd 11239 . . . . . . . . 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 11237 . . . . . . . . 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 7419 . . . . . . . . . . . . 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 3683 . . . . . . . . . . 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 25840 . . . . . . . . 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 12221 . . . . . . . . . . 11 (𝑅 ∈ β„• β†’ 𝑅 ∈ ℝ)
102 nnrp 12987 . . . . . . . . . . 11 ((π‘β€˜π‘š) ∈ β„• β†’ (π‘β€˜π‘š) ∈ ℝ+)
103 mod0 13843 . . . . . . . . . . 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 12674 . . . . . . 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 25723 . . . . 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 4790 . . . . . . 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 7418 . . . . . . 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 7086 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (πΉβ€˜π‘§) ∈ β„‚)
11653nnne0d 12264 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑅 β‰  0)
117116adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 𝑅 β‰  0)
118115, 55, 117divcan3d 11997 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ ((𝑅 Β· (πΉβ€˜π‘§)) / 𝑅) = (πΉβ€˜π‘§))
119118mpteq2dva 5248 . . . . . . . . 9 (πœ‘ β†’ (𝑧 ∈ β„‚ ↦ ((𝑅 Β· (πΉβ€˜π‘§)) / 𝑅)) = (𝑧 ∈ β„‚ ↦ (πΉβ€˜π‘§)))
120 ovexd 7446 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ (𝑅 Β· (πΉβ€˜π‘§)) ∈ V)
1213, 120, 6, 15, 9offval2 7692 . . . . . . . . 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 11991 . . . . . . . . . 10 (πœ‘ β†’ (0 / 𝑅) = 0)
124123mpteq2dv 5250 . . . . . . . . 9 (πœ‘ β†’ (𝑧 ∈ β„‚ ↦ (0 / 𝑅)) = (𝑧 ∈ β„‚ ↦ 0))
125 0cnd 11209 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ β„‚) β†’ 0 ∈ β„‚)
126 df-0p 25194 . . . . . . . . . . . 12 0𝑝 = (β„‚ Γ— {0})
127 fconstmpt 5738 . . . . . . . . . . . 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 7692 . . . . . . . . 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 4790 . . . 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 6777 . . . . . . 7 (β„‚ Γ— {𝑅}):β„‚βŸΆ{𝑅}
139 ffn 6717 . . . . . . 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 6718 . . . . . 6 (πœ‘ β†’ 𝐹 Fn β„‚)
142 inidm 4218 . . . . . 6 (β„‚ ∩ β„‚) = β„‚
1435fvconst2 7207 . . . . . . 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 7683 . . . . 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 11415 . . . 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 6890 . . . . 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 25836 . 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 3945   βŠ† wss 3948  βˆ…c0 4322  {csn 4628   ↦ cmpt 5231   Γ— cxp 5674   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413   ∘f cof 7670  infcinf 9438  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   Β· cmul 11117   < clt 11250   / cdiv 11873  β„•cn 12214  β„•0cn0 12474  β„€cz 12560  β„€β‰₯cuz 12824  β„šcq 12934  β„+crp 12976  ...cfz 13486   mod cmo 13836  seqcseq 13968  β†‘cexp 14029  Ξ£csu 15634  0𝑝c0p 25193  Polycply 25705  coeffccoe 25707  degcdgr 25708  π”Έcaa 25834
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
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-op 4635  df-uni 4909  df-int 4951  df-iun 4999  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-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  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-n0 12475  df-z 12561  df-uz 12825  df-q 12935  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-rlim 15435  df-sum 15635  df-0p 25194  df-ply 25709  df-coe 25711  df-dgr 25712  df-aa 25835
This theorem is referenced by:  elqaa  25842
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