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Theorem diophrex 41498
Description: Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
diophrex ((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
Distinct variable groups:   𝑑,𝑁,𝑒   𝑑,𝑆,𝑒
Allowed substitution hints:   𝑀(𝑒,𝑑)
Proof of Theorem diophrex
Dummy variables π‘Ž 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2736 . . . . 5 (π‘Ž = 𝑑 β†’ (π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ 𝑑 = (𝑏 β†Ύ (1...𝑁))))
21rexbidv 3178 . . . 4 (π‘Ž = 𝑑 β†’ (βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘ ∈ 𝑆 𝑑 = (𝑏 β†Ύ (1...𝑁))))
3 reseq1 5973 . . . . . 6 (𝑏 = 𝑒 β†’ (𝑏 β†Ύ (1...𝑁)) = (𝑒 β†Ύ (1...𝑁)))
43eqeq2d 2743 . . . . 5 (𝑏 = 𝑒 β†’ (𝑑 = (𝑏 β†Ύ (1...𝑁)) ↔ 𝑑 = (𝑒 β†Ύ (1...𝑁))))
54cbvrexvw 3235 . . . 4 (βˆƒπ‘ ∈ 𝑆 𝑑 = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁)))
62, 5bitrdi 286 . . 3 (π‘Ž = 𝑑 β†’ (βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁))))
76cbvabv 2805 . 2 {π‘Ž ∣ βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁))} = {𝑑 ∣ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁))}
8 rexeq 3321 . . . . . 6 (𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)} β†’ (βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁))))
98abbidv 2801 . . . . 5 (𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)} β†’ {π‘Ž ∣ βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁))} = {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁))})
109adantl 482 . . . 4 ((((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) ∧ 𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁))} = {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁))})
11 eqeq1 2736 . . . . . . . . . . 11 (𝑑 = 𝑏 β†’ (𝑑 = (𝑒 β†Ύ (1...𝑀)) ↔ 𝑏 = (𝑒 β†Ύ (1...𝑀))))
1211anbi1d 630 . . . . . . . . . 10 (𝑑 = 𝑏 β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ↔ (𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)))
1312rexbidv 3178 . . . . . . . . 9 (𝑑 = 𝑏 β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)))
1413rexab 3689 . . . . . . . 8 (βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘(βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))))
15 r19.41v 3188 . . . . . . . . . 10 (βˆƒπ‘’ ∈ (β„•0 ↑m β„•)((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ (βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))))
1615exbii 1850 . . . . . . . . 9 (βˆƒπ‘βˆƒπ‘’ ∈ (β„•0 ↑m β„•)((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ βˆƒπ‘(βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))))
17 rexcom4 3285 . . . . . . . . . 10 (βˆƒπ‘’ ∈ (β„•0 ↑m β„•)βˆƒπ‘((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ βˆƒπ‘βˆƒπ‘’ ∈ (β„•0 ↑m β„•)((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))))
18 anass 469 . . . . . . . . . . . . . 14 (((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ (𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ ((π‘β€˜π‘’) = 0 ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))))
1918exbii 1850 . . . . . . . . . . . . 13 (βˆƒπ‘((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ βˆƒπ‘(𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ ((π‘β€˜π‘’) = 0 ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))))
20 vex 3478 . . . . . . . . . . . . . . 15 𝑒 ∈ V
2120resex 6027 . . . . . . . . . . . . . 14 (𝑒 β†Ύ (1...𝑀)) ∈ V
22 reseq1 5973 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑒 β†Ύ (1...𝑀)) β†’ (𝑏 β†Ύ (1...𝑁)) = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)))
2322eqeq2d 2743 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒 β†Ύ (1...𝑀)) β†’ (π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁))))
2423anbi2d 629 . . . . . . . . . . . . . 14 (𝑏 = (𝑒 β†Ύ (1...𝑀)) β†’ (((π‘β€˜π‘’) = 0 ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ ((π‘β€˜π‘’) = 0 ∧ π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)))))
2521, 24ceqsexv 3525 . . . . . . . . . . . . 13 (βˆƒπ‘(𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ ((π‘β€˜π‘’) = 0 ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) ↔ ((π‘β€˜π‘’) = 0 ∧ π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁))))
2619, 25bitri 274 . . . . . . . . . . . 12 (βˆƒπ‘((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ ((π‘β€˜π‘’) = 0 ∧ π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁))))
27 ancom 461 . . . . . . . . . . . . 13 (((π‘β€˜π‘’) = 0 ∧ π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁))) ↔ (π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0))
28 simpl2 1192 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘))
29 fzss2 13537 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (β„€β‰₯β€˜π‘) β†’ (1...𝑁) βŠ† (1...𝑀))
30 resabs1 6009 . . . . . . . . . . . . . . . 16 ((1...𝑁) βŠ† (1...𝑀) β†’ ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) = (𝑒 β†Ύ (1...𝑁)))
3128, 29, 303syl 18 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) = (𝑒 β†Ύ (1...𝑁)))
3231eqeq2d 2743 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ (π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) ↔ π‘Ž = (𝑒 β†Ύ (1...𝑁))))
3332anbi1d 630 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ ((π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ (π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
3427, 33bitrid 282 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ (((π‘β€˜π‘’) = 0 ∧ π‘Ž = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁))) ↔ (π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
3526, 34bitrid 282 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ (βˆƒπ‘((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ (π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
3635rexbidv 3178 . . . . . . . . . 10 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m β„•)βˆƒπ‘((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
3717, 36bitr3id 284 . . . . . . . . 9 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ (βˆƒπ‘βˆƒπ‘’ ∈ (β„•0 ↑m β„•)((𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
3816, 37bitr3id 284 . . . . . . . 8 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ (βˆƒπ‘(βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑏 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0) ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
3914, 38bitrid 282 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ (βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)))
4039abbidv 2801 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁))} = {π‘Ž ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
41 eldioph3 41489 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ {π‘Ž ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
42413ad2antl1 1185 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ {π‘Ž ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(π‘Ž = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
4340, 42eqeltrd 2833 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
4443adantr 481 . . . 4 ((((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) ∧ 𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}π‘Ž = (𝑏 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
4510, 44eqeltrd 2833 . . 3 ((((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) ∧ 𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
46 eldioph3b 41488 . . . . 5 (𝑆 ∈ (Diophβ€˜π‘€) ↔ (𝑀 ∈ β„•0 ∧ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)}))
4746simprbi 497 . . . 4 (𝑆 ∈ (Diophβ€˜π‘€) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)})
48473ad2ant3 1135 . . 3 ((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜β„•)𝑆 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m β„•)(𝑑 = (𝑒 β†Ύ (1...𝑀)) ∧ (π‘β€˜π‘’) = 0)})
4945, 48r19.29a 3162 . 2 ((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
507, 49eqeltrrid 2838 1 ((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆƒwrex 3070   βŠ† wss 3947   β†Ύ cres 5677  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  0cc0 11106  1c1 11107  β„•cn 12208  β„•0cn0 12468  β„€β‰₯cuz 12818  ...cfz 13480  mzPolycmzp 41445  Diophcdioph 41478
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
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-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-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-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-oadd 8466  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  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-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287  df-mzpcl 41446  df-mzp 41447  df-dioph 41479
This theorem is referenced by:  rexrabdioph  41517
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