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Theorem eldioph2b 42248
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 42257 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 42242 . . 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 6788 . . . . . . . . . 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 42234 . . . . . . . . 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 1368 . . . . . . . 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 42244 . . . . . . . . . 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 2731 . . . . . . . . 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 1368 . . . . . . . 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 6891 . . . . . . . . . . . . 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 2727 . . . . . . . . . . . 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 628 . . . . . . . . . . 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 3169 . . . . . . . . . 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 2794 . . . . . . . . 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 3623 . . . . . . . 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 582 . . . . . . 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 42246 . . . . . . . . 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 3489 . . . . . . . 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 3152 . . . . . 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 2729 . . . . . . 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 3169 . . . . . 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 3202 . . . 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 489 . . 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 483 . . . 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 483 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ 𝑆 ∈ V) ∧ (Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) βŠ† 𝑆)) ∧ 𝑝 ∈ (mzPolyβ€˜π‘†)) β†’ 𝑝 ∈ (mzPolyβ€˜π‘†))
41 eldioph2 42247 . . . . . 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 1372 . . . . 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 479 . . . 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 2825 . . 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 3147 . 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 394   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2702  βˆƒwrex 3060  Vcvv 3463   βŠ† wss 3939   ↦ cmpt 5226   I cid 5569   β†Ύ cres 5674   ∘ ccom 5676  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543  (class class class)co 7416   ↑m cmap 8843  Fincfn 8962  0cc0 11138  1c1 11139  β„•0cn0 12502  β„€cz 12588  β„€β‰₯cuz 12852  ...cfz 13516  mzPolycmzp 42207  Diophcdioph 42240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-hash 14322  df-mzpcl 42208  df-mzp 42209  df-dioph 42241
This theorem is referenced by:  eldioph3b  42250  diophin  42257  diophun  42258  eldioph4b  42296
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