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Theorem diophrex 41816
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 2735 . . . . 5 (π‘Ž = 𝑑 β†’ (π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ 𝑑 = (𝑏 β†Ύ (1...𝑁))))
21rexbidv 3177 . . . 4 (π‘Ž = 𝑑 β†’ (βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘ ∈ 𝑆 𝑑 = (𝑏 β†Ύ (1...𝑁))))
3 reseq1 5975 . . . . . 6 (𝑏 = 𝑒 β†’ (𝑏 β†Ύ (1...𝑁)) = (𝑒 β†Ύ (1...𝑁)))
43eqeq2d 2742 . . . . 5 (𝑏 = 𝑒 β†’ (𝑑 = (𝑏 β†Ύ (1...𝑁)) ↔ 𝑑 = (𝑒 β†Ύ (1...𝑁))))
54cbvrexvw 3234 . . . 4 (βˆƒπ‘ ∈ 𝑆 𝑑 = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁)))
62, 5bitrdi 287 . . 3 (π‘Ž = 𝑑 β†’ (βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁))))
76cbvabv 2804 . 2 {π‘Ž ∣ βˆƒπ‘ ∈ 𝑆 π‘Ž = (𝑏 β†Ύ (1...𝑁))} = {𝑑 ∣ βˆƒπ‘’ ∈ 𝑆 𝑑 = (𝑒 β†Ύ (1...𝑁))}
8 rexeq 3320 . . . . . 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 2800 . . . . 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 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‘ ∈ 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 2735 . . . . . . . . . . 11 (𝑑 = 𝑏 β†’ (𝑑 = (𝑒 β†Ύ (1...𝑀)) ↔ 𝑏 = (𝑒 β†Ύ (1...𝑀))))
1211anbi1d 629 . . . . . . . . . 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 3177 . . . . . . . . 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 3690 . . . . . . . 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 3187 . . . . . . . . . 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 1849 . . . . . . . . 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 3284 . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 1849 . . . . . . . . . . . . 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 3477 . . . . . . . . . . . . . . 15 𝑒 ∈ V
2120resex 6029 . . . . . . . . . . . . . 14 (𝑒 β†Ύ (1...𝑀)) ∈ V
22 reseq1 5975 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑒 β†Ύ (1...𝑀)) β†’ (𝑏 β†Ύ (1...𝑁)) = ((𝑒 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)))
2322eqeq2d 2742 . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . 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 275 . . . . . . . . . . . 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 460 . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑆 ∈ (Diophβ€˜π‘€)) ∧ 𝑐 ∈ (mzPolyβ€˜β„•)) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘))
29 fzss2 13546 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (β„€β‰₯β€˜π‘) β†’ (1...𝑁) βŠ† (1...𝑀))
30 resabs1 6011 . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . 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 283 . . . . . . . . . . . 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 283 . . . . . . . . . . 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 3177 . . . . . . . . . 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 285 . . . . . . . . 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 285 . . . . . . . 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 283 . . . . . . 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 2800 . . . . . 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 41807 . . . . . . 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 1184 . . . . . 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 2832 . . . . 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 480 . . . 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 2832 . . 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 41806 . . . . 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 496 . . . 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 1134 . . 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 3161 . 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 2837 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 395   ∧ w3a 1086   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105  {cab 2708  βˆƒwrex 3069   βŠ† wss 3948   β†Ύ cres 5678  β€˜cfv 6543  (class class class)co 7412   ↑m cmap 8823  0cc0 11113  1c1 11114  β„•cn 12217  β„•0cn0 12477  β„€β‰₯cuz 12827  ...cfz 13489  mzPolycmzp 41763  Diophcdioph 41796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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-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-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-oadd 8473  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9899  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-hash 14296  df-mzpcl 41764  df-mzp 41765  df-dioph 41797
This theorem is referenced by:  rexrabdioph  41835
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