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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldioph2b Structured version   Visualization version   GIF version

Theorem eldioph2b 41486
Description: While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set (𝑆 βˆ– (1...𝑁)). For instance, in diophin 41495 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
eldioph2b (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) β†’ (𝐴 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
Distinct variable groups:   𝐴,𝑝   𝑒,𝑁,𝑑,𝑝   𝑒,𝑆,𝑑,𝑝
Allowed substitution hints:   𝐴(𝑒,𝑑)

Proof of Theorem eldioph2b
Dummy variables π‘Ž 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldiophb 41480 . . 3 (𝐴 ∈ (Diophβ€˜π‘) ↔ (𝑁 ∈ β„•0 ∧ βˆƒπ‘Ž ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘Ž))𝐴 = {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}))
2 simp-5r 784 . . . . . . . . 9 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑆 ∈ V)
3 simprr 771 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))
43ad2antrr 724 . . . . . . . . 9 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))
5 simprl 769 . . . . . . . . . 10 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑐:(1...π‘Ž)–1-1→𝑆)
6 f1f 6784 . . . . . . . . . 10 (𝑐:(1...π‘Ž)–1-1→𝑆 β†’ 𝑐:(1...π‘Ž)βŸΆπ‘†)
75, 6syl 17 . . . . . . . . 9 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑐:(1...π‘Ž)βŸΆπ‘†)
8 mzprename 41472 . . . . . . . . 9 ((𝑆 ∈ V ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)) ∧ 𝑐:(1...π‘Ž)βŸΆπ‘†) β†’ (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) ∈ (mzPolyβ€˜π‘†))
92, 4, 7, 8syl3anc 1371 . . . . . . . 8 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) ∈ (mzPolyβ€˜π‘†))
10 simprr 771 . . . . . . . . 9 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))
11 diophrw 41482 . . . . . . . . . 10 ((𝑆 ∈ V ∧ 𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)})
1211eqcomd 2738 . . . . . . . . 9 ((𝑆 ∈ V ∧ 𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))) β†’ {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0)})
132, 5, 10, 12syl3anc 1371 . . . . . . . 8 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0)})
14 fveq1 6887 . . . . . . . . . . . . 13 (𝑝 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) β†’ (π‘β€˜π‘’) = ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’))
1514eqeq1d 2734 . . . . . . . . . . . 12 (𝑝 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) β†’ ((π‘β€˜π‘’) = 0 ↔ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0))
1615anbi2d 629 . . . . . . . . . . 11 (𝑝 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ (𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0)))
1716rexbidv 3178 . . . . . . . . . 10 (𝑝 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0)))
1817abbidv 2801 . . . . . . . . 9 (𝑝 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0)})
1918rspceeqv 3632 . . . . . . . 8 (((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐))) ∈ (mzPolyβ€˜π‘†) ∧ {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 ∘ 𝑐)))β€˜π‘’) = 0)}) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†){𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
209, 13, 19syl2anc 584 . . . . . . 7 ((((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†){𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
21 simplll 773 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ 𝑁 ∈ β„•0)
22 simplrl 775 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ Β¬ 𝑆 ∈ Fin)
23 simplrr 776 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ (1...𝑁) βŠ† 𝑆)
24 simprl 769 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ π‘Ž ∈ (β„€β‰₯β€˜π‘))
25 eldioph2lem2 41484 . . . . . . . . 9 (((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ π‘Ž ∈ (β„€β‰₯β€˜π‘))) β†’ βˆƒπ‘(𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
2621, 22, 23, 24, 25syl22anc 837 . . . . . . . 8 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ βˆƒπ‘(𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
27 rexv 3499 . . . . . . . 8 (βˆƒπ‘ ∈ V (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))) ↔ βˆƒπ‘(𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
2826, 27sylibr 233 . . . . . . 7 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ βˆƒπ‘ ∈ V (𝑐:(1...π‘Ž)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
2920, 28r19.29a 3162 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†){𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
30 eqeq1 2736 . . . . . . 7 (𝐴 = {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} β†’ (𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ↔ {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
3130rexbidv 3178 . . . . . 6 (𝐴 = {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} β†’ (βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ↔ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†){𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
3229, 31syl5ibrcom 246 . . . . 5 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ (β„€β‰₯β€˜π‘) ∧ 𝑏 ∈ (mzPolyβ€˜(1...π‘Ž)))) β†’ (𝐴 = {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} β†’ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
3332rexlimdvva 3211 . . . 4 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) β†’ (βˆƒπ‘Ž ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘Ž))𝐴 = {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)} β†’ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
3433adantld 491 . . 3 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) β†’ ((𝑁 ∈ β„•0 ∧ βˆƒπ‘Ž ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘ ∈ (mzPolyβ€˜(1...π‘Ž))𝐴 = {𝑑 ∣ βˆƒπ‘‘ ∈ (β„•0 ↑m (1...π‘Ž))(𝑑 = (𝑑 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘‘) = 0)}) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
351, 34biimtrid 241 . 2 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) β†’ (𝐴 ∈ (Diophβ€˜π‘) β†’ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
36 simpr 485 . . . 4 (((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) ∧ 𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) β†’ 𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)})
37 simplll 773 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) β†’ 𝑁 ∈ β„•0)
38 simpllr 774 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) β†’ 𝑆 ∈ V)
39 simplrr 776 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) β†’ (1...𝑁) βŠ† 𝑆)
40 simpr 485 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) β†’ 𝑝 ∈ (mzPolyβ€˜π‘†))
41 eldioph2 41485 . . . . . 6 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
4237, 38, 39, 40, 41syl121anc 1375 . . . . 5 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
4342adantr 481 . . . 4 (((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) ∧ 𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
4436, 43eqeltrd 2833 . . 3 (((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) ∧ 𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}) β†’ 𝐴 ∈ (Diophβ€˜π‘))
4544rexlimdva2 3157 . 2 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) β†’ (βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)} β†’ 𝐴 ∈ (Diophβ€˜π‘)))
4635, 45impbid 211 1 (((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) β†’ (𝐴 ∈ (Diophβ€˜π‘) ↔ βˆƒπ‘ ∈ (mzPolyβ€˜π‘†)𝐴 = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘β€˜π‘’) = 0)}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3947   ↦ cmpt 5230   I cid 5572   β†Ύ cres 5677   ∘ ccom 5679  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  Fincfn 8935  0cc0 11106  1c1 11107  β„•0cn0 12468  β„€cz 12554  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-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:  eldioph3b  41488  diophin  41495  diophun  41496  eldioph4b  41534
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