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Theorem eldioph2 41485
Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 41475. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
Assertion
Ref Expression
eldioph2 ((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‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
Distinct variable groups:   𝑑,𝑃,𝑒   𝑑,𝑆,𝑒   𝑑,𝑁,𝑒

Proof of Theorem eldioph2
Dummy variables π‘Ž 𝑏 𝑐 𝑒 𝑔 β„Ž 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mzpcompact2 41475 . . 3 (𝑃 ∈ (mzPolyβ€˜π‘†) β†’ βˆƒπ‘Ž ∈ Fin βˆƒπ‘ ∈ (mzPolyβ€˜π‘Ž)(π‘Ž βŠ† 𝑆 ∧ 𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))))
213ad2ant3 1135 . 2 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) β†’ βˆƒπ‘Ž ∈ Fin βˆƒπ‘ ∈ (mzPolyβ€˜π‘Ž)(π‘Ž βŠ† 𝑆 ∧ 𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))))
3 fveq1 6887 . . . . . . . . . 10 (𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))) β†’ (π‘ƒβ€˜π‘’) = ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’))
43eqeq1d 2734 . . . . . . . . 9 (𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))) β†’ ((π‘ƒβ€˜π‘’) = 0 ↔ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0))
54anbi2d 629 . . . . . . . 8 (𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))) β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0) ↔ (𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)))
65rexbidv 3178 . . . . . . 7 (𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))) β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)))
76abbidv 2801 . . . . . 6 (𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)})
87ad2antll 727 . . . . 5 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ (π‘Ž βŠ† 𝑆 ∧ 𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)})
9 simplll 773 . . . . . . . . . . 11 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ 𝑁 ∈ β„•0)
10 simplrl 775 . . . . . . . . . . . 12 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ π‘Ž ∈ Fin)
11 fzfi 13933 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
12 unfi 9168 . . . . . . . . . . . 12 ((π‘Ž ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ (π‘Ž βˆͺ (1...𝑁)) ∈ Fin)
1310, 11, 12sylancl 586 . . . . . . . . . . 11 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ (π‘Ž βˆͺ (1...𝑁)) ∈ Fin)
14 ssun2 4172 . . . . . . . . . . . 12 (1...𝑁) βŠ† (π‘Ž βˆͺ (1...𝑁))
1514a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ (1...𝑁) βŠ† (π‘Ž βˆͺ (1...𝑁)))
16 eldioph2lem1 41483 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ (π‘Ž βˆͺ (1...𝑁)) ∈ Fin ∧ (1...𝑁) βŠ† (π‘Ž βˆͺ (1...𝑁))) β†’ βˆƒπ‘ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘‘ ∈ V (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
179, 13, 15, 16syl3anc 1371 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ βˆƒπ‘ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘‘ ∈ V (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
18 f1ococnv2 6857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ (𝑑 ∘ ◑𝑑) = ( I β†Ύ (π‘Ž βˆͺ (1...𝑁))))
1918ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (𝑑 ∘ ◑𝑑) = ( I β†Ύ (π‘Ž βˆͺ (1...𝑁))))
2019reseq1d 5978 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ ((𝑑 ∘ ◑𝑑) β†Ύ π‘Ž) = (( I β†Ύ (π‘Ž βˆͺ (1...𝑁))) β†Ύ π‘Ž))
21 ssun1 4171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 π‘Ž βŠ† (π‘Ž βˆͺ (1...𝑁))
22 resabs1 6009 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘Ž βŠ† (π‘Ž βˆͺ (1...𝑁)) β†’ (( I β†Ύ (π‘Ž βˆͺ (1...𝑁))) β†Ύ π‘Ž) = ( I β†Ύ π‘Ž))
2321, 22ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (( I β†Ύ (π‘Ž βˆͺ (1...𝑁))) β†Ύ π‘Ž) = ( I β†Ύ π‘Ž)
2420, 23eqtr2di 2789 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ ( I β†Ύ π‘Ž) = ((𝑑 ∘ ◑𝑑) β†Ύ π‘Ž))
25 resco 6246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∘ ◑𝑑) β†Ύ π‘Ž) = (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž))
2624, 25eqtrdi 2788 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ ( I β†Ύ π‘Ž) = (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž)))
2726adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ ( I β†Ύ π‘Ž) = (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž)))
2827coeq2d 5860 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ (𝑒 ∘ ( I β†Ύ π‘Ž)) = (𝑒 ∘ (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž))))
29 coires1 6260 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ ( I β†Ύ π‘Ž)) = (𝑒 β†Ύ π‘Ž)
30 coass 6261 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž)) = (𝑒 ∘ (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž)))
3130eqcomi 2741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž))) = ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))
3228, 29, 313eqtr3g 2795 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ (𝑒 β†Ύ π‘Ž) = ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž)))
3332fveq2d 6892 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ (π‘β€˜(𝑒 β†Ύ π‘Ž)) = (π‘β€˜((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))))
34 ovexd 7440 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ (1...𝑐) ∈ V)
35 simpr 485 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ 𝑒 ∈ (β„€ ↑m 𝑆))
36 f1of1 6829 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ 𝑑:(1...𝑐)–1-1β†’(π‘Ž βˆͺ (1...𝑁)))
3736ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑑:(1...𝑐)–1-1β†’(π‘Ž βˆͺ (1...𝑁)))
38 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ π‘Ž βŠ† 𝑆)
39 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) β†’ (1...𝑁) βŠ† 𝑆)
4039ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ (1...𝑁) βŠ† 𝑆)
4138, 40unssd 4185 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ (π‘Ž βˆͺ (1...𝑁)) βŠ† 𝑆)
4241ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (π‘Ž βˆͺ (1...𝑁)) βŠ† 𝑆)
43 f1ss 6790 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑:(1...𝑐)–1-1β†’(π‘Ž βˆͺ (1...𝑁)) ∧ (π‘Ž βˆͺ (1...𝑁)) βŠ† 𝑆) β†’ 𝑑:(1...𝑐)–1-1→𝑆)
4437, 42, 43syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑑:(1...𝑐)–1-1→𝑆)
45 f1f 6784 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑:(1...𝑐)–1-1→𝑆 β†’ 𝑑:(1...𝑐)βŸΆπ‘†)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑑:(1...𝑐)βŸΆπ‘†)
4746adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ 𝑑:(1...𝑐)βŸΆπ‘†)
48 mapco2g 41437 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑐) ∈ V ∧ 𝑒 ∈ (β„€ ↑m 𝑆) ∧ 𝑑:(1...𝑐)βŸΆπ‘†) β†’ (𝑒 ∘ 𝑑) ∈ (β„€ ↑m (1...𝑐)))
4934, 35, 47, 48syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ (𝑒 ∘ 𝑑) ∈ (β„€ ↑m (1...𝑐)))
50 coeq1 5855 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„Ž = (𝑒 ∘ 𝑑) β†’ (β„Ž ∘ (◑𝑑 β†Ύ π‘Ž)) = ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž)))
5150fveq2d 6892 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž = (𝑒 ∘ 𝑑) β†’ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))) = (π‘β€˜((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))))
52 eqid 2732 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž)))) = (β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))
53 fvex 6901 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘β€˜((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))) ∈ V
5451, 52, 53fvmpt 6995 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑒 ∘ 𝑑) ∈ (β„€ ↑m (1...𝑐)) β†’ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)) = (π‘β€˜((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))))
5549, 54syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)) = (π‘β€˜((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))))
5633, 55eqtr4d 2775 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) ∧ 𝑒 ∈ (β„€ ↑m 𝑆)) β†’ (π‘β€˜(𝑒 β†Ύ π‘Ž)) = ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))
5756mpteq2dva 5247 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))) = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑))))
5857fveq1d 6890 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))β€˜π‘’))
5958eqeq1d 2734 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0 ↔ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))β€˜π‘’) = 0))
6059anbi2d 629 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ ((𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0) ↔ (𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))β€˜π‘’) = 0)))
6160rexbidv 3178 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0) ↔ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))β€˜π‘’) = 0)))
6261abbidv 2801 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 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 β†Ύ π‘Ž)))β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))β€˜π‘’) = 0)})
63 simplrl 775 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) β†’ 𝑆 ∈ V)
6463ad3antrrr 728 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑆 ∈ V)
65 simprr 771 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))
66 diophrw 41482 . . . . . . . . . . . . . . 15 ((𝑆 ∈ V ∧ 𝑑:(1...𝑐)–1-1→𝑆 ∧ (𝑑 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) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘” ∈ (β„•0 ↑m (1...𝑐))(𝑑 = (𝑔 β†Ύ (1...𝑁)) ∧ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜π‘”) = 0)})
6764, 44, 65, 66syl3anc 1371 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 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) ↦ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜(𝑒 ∘ 𝑑)))β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘” ∈ (β„•0 ↑m (1...𝑐))(𝑑 = (𝑔 β†Ύ (1...𝑁)) ∧ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜π‘”) = 0)})
6862, 67eqtrd 2772 . . . . . . . . . . . . 13 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 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 β†Ύ π‘Ž)))β€˜π‘’) = 0)} = {𝑑 ∣ βˆƒπ‘” ∈ (β„•0 ↑m (1...𝑐))(𝑑 = (𝑔 β†Ύ (1...𝑁)) ∧ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜π‘”) = 0)})
69 simp-5l 783 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑁 ∈ β„•0)
70 simplrl 775 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑐 ∈ (β„€β‰₯β€˜π‘))
71 ovexd 7440 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (1...𝑐) ∈ V)
72 simplrr 776 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ 𝑏 ∈ (mzPolyβ€˜π‘Ž))
7372ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ 𝑏 ∈ (mzPolyβ€˜π‘Ž))
74 f1ocnv 6842 . . . . . . . . . . . . . . . . . 18 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ ◑𝑑:(π‘Ž βˆͺ (1...𝑁))–1-1-ontoβ†’(1...𝑐))
75 f1of 6830 . . . . . . . . . . . . . . . . . 18 (◑𝑑:(π‘Ž βˆͺ (1...𝑁))–1-1-ontoβ†’(1...𝑐) β†’ ◑𝑑:(π‘Ž βˆͺ (1...𝑁))⟢(1...𝑐))
7674, 75syl 17 . . . . . . . . . . . . . . . . 17 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ ◑𝑑:(π‘Ž βˆͺ (1...𝑁))⟢(1...𝑐))
77 fssres 6754 . . . . . . . . . . . . . . . . 17 ((◑𝑑:(π‘Ž βˆͺ (1...𝑁))⟢(1...𝑐) ∧ π‘Ž βŠ† (π‘Ž βˆͺ (1...𝑁))) β†’ (◑𝑑 β†Ύ π‘Ž):π‘ŽβŸΆ(1...𝑐))
7876, 21, 77sylancl 586 . . . . . . . . . . . . . . . 16 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ (◑𝑑 β†Ύ π‘Ž):π‘ŽβŸΆ(1...𝑐))
7978ad2antrl 726 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (◑𝑑 β†Ύ π‘Ž):π‘ŽβŸΆ(1...𝑐))
80 mzprename 41472 . . . . . . . . . . . . . . 15 (((1...𝑐) ∈ V ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž) ∧ (◑𝑑 β†Ύ π‘Ž):π‘ŽβŸΆ(1...𝑐)) β†’ (β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž)))) ∈ (mzPolyβ€˜(1...𝑐)))
8171, 73, 79, 80syl3anc 1371 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ (β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž)))) ∈ (mzPolyβ€˜(1...𝑐)))
82 eldioph 41481 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„•0 ∧ 𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ (β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž)))) ∈ (mzPolyβ€˜(1...𝑐))) β†’ {𝑑 ∣ βˆƒπ‘” ∈ (β„•0 ↑m (1...𝑐))(𝑑 = (𝑔 β†Ύ (1...𝑁)) ∧ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜π‘”) = 0)} ∈ (Diophβ€˜π‘))
8369, 70, 81, 82syl3anc 1371 . . . . . . . . . . . . 13 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))) β†’ {𝑑 ∣ βˆƒπ‘” ∈ (β„•0 ↑m (1...𝑐))(𝑑 = (𝑔 β†Ύ (1...𝑁)) ∧ ((β„Ž ∈ (β„€ ↑m (1...𝑐)) ↦ (π‘β€˜(β„Ž ∘ (◑𝑑 β†Ύ π‘Ž))))β€˜π‘”) = 0)} ∈ (Diophβ€˜π‘))
8468, 83eqeltrd 2833 . . . . . . . . . . . 12 ((((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) ∧ (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 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 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
8584ex 413 . . . . . . . . . . 11 (((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) ∧ (𝑐 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑑 ∈ V)) β†’ ((𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 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 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘)))
8685rexlimdvva 3211 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ (βˆƒπ‘ ∈ (β„€β‰₯β€˜π‘)βˆƒπ‘‘ ∈ V (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) ∧ (𝑑 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 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘)))
8717, 86mpd 15 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
8887exp31 420 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) β†’ ((π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž)) β†’ (π‘Ž βŠ† 𝑆 β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))))
89883adant3 1132 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) β†’ ((π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž)) β†’ (π‘Ž βŠ† 𝑆 β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))))
9089imp31 418 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ π‘Ž βŠ† 𝑆) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
9190adantrr 715 . . . . 5 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ (π‘Ž βŠ† 𝑆 ∧ 𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ ((𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))β€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
928, 91eqeltrd 2833 . . . 4 ((((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) ∧ (π‘Ž βŠ† 𝑆 ∧ 𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž))))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
9392ex 413 . . 3 (((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) β†’ ((π‘Ž βŠ† 𝑆 ∧ 𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘)))
9493rexlimdvva 3211 . 2 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆) ∧ 𝑃 ∈ (mzPolyβ€˜π‘†)) β†’ (βˆƒπ‘Ž ∈ Fin βˆƒπ‘ ∈ (mzPolyβ€˜π‘Ž)(π‘Ž βŠ† 𝑆 ∧ 𝑃 = (𝑒 ∈ (β„€ ↑m 𝑆) ↦ (π‘β€˜(𝑒 β†Ύ π‘Ž)))) β†’ {𝑑 ∣ βˆƒπ‘’ ∈ (β„•0 ↑m 𝑆)(𝑑 = (𝑒 β†Ύ (1...𝑁)) ∧ (π‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘)))
952, 94mpd 15 1 ((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‘ƒβ€˜π‘’) = 0)} ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆƒwrex 3070  Vcvv 3474   βˆͺ cun 3945   βŠ† wss 3947   ↦ cmpt 5230   I cid 5572  β—‘ccnv 5674   β†Ύ cres 5677   ∘ ccom 5679  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539  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:  eldioph2b  41486  diophin  41495  diophun  41496
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