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Theorem eldioph2 41114
Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 41104. (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 41104 . . 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 1136 . 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 6846 . . . . . . . . . 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 2739 . . . . . . . . 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 630 . . . . . . . 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 3176 . . . . . . 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 2806 . . . . . 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 728 . . . . 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 774 . . . . . . . . . . 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 776 . . . . . . . . . . . 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 13884 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
12 unfi 9123 . . . . . . . . . . . 12 ((π‘Ž ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ (π‘Ž βˆͺ (1...𝑁)) ∈ Fin)
1310, 11, 12sylancl 587 . . . . . . . . . . 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 4138 . . . . . . . . . . . 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 41112 . . . . . . . . . . 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 1372 . . . . . . . . . 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 6816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ (𝑑 ∘ ◑𝑑) = ( I β†Ύ (π‘Ž βˆͺ (1...𝑁))))
1918ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5941 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4137 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 π‘Ž βŠ† (π‘Ž βˆͺ (1...𝑁))
22 resabs1 5972 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6207 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∘ ◑𝑑) β†Ύ π‘Ž) = (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž))
2624, 25eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . 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 5823 . . . . . . . . . . . . . . . . . . . . . . 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 6221 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ ( I β†Ύ π‘Ž)) = (𝑒 β†Ύ π‘Ž)
30 coass 6222 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž)) = (𝑒 ∘ (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž)))
3130eqcomi 2746 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž))) = ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))
3228, 29, 313eqtr3g 2800 . . . . . . . . . . . . . . . . . . . . . 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 6851 . . . . . . . . . . . . . . . . . . . . 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 7397 . . . . . . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . . . . . 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 6788 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ 𝑑:(1...𝑐)–1-1β†’(π‘Ž βˆͺ (1...𝑁)))
3736ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) β†’ (1...𝑁) βŠ† 𝑆)
4039ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4151 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6749 . . . . . . . . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . . . . . . . . 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 6743 . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . 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 41066 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑐) ∈ V ∧ 𝑒 ∈ (β„€ ↑m 𝑆) ∧ 𝑑:(1...𝑐)βŸΆπ‘†) β†’ (𝑒 ∘ 𝑑) ∈ (β„€ ↑m (1...𝑐)))
4934, 35, 47, 48syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . 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 5818 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„Ž = (𝑒 ∘ 𝑑) β†’ (β„Ž ∘ (◑𝑑 β†Ύ π‘Ž)) = ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž)))
5150fveq2d 6851 . . . . . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . . . 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 6860 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘β€˜((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))) ∈ V
5451, 52, 53fvmpt 6953 . . . . . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . . . . . 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 5210 . . . . . . . . . . . . . . . . . . 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 6849 . . . . . . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) β†’ 𝑆 ∈ V)
6463ad3antrrr 729 . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . 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 41111 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 7397 . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . 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 6801 . . . . . . . . . . . . . . . . . 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 6789 . . . . . . . . . . . . . . . . . 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 6713 . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . 16 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ (◑𝑑 β†Ύ π‘Ž):π‘ŽβŸΆ(1...𝑐))
7978ad2antrl 727 . . . . . . . . . . . . . . 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 41101 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 41110 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 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 2838 . . . . . . . . . . . 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 414 . . . . . . . . . . 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 3206 . . . . . . . . . 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 421 . . . . . . . 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 1133 . . . . . . 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 419 . . . . . 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 716 . . . . 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 2838 . . . 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 414 . . 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 3206 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆƒwrex 3074  Vcvv 3448   βˆͺ cun 3913   βŠ† wss 3915   ↦ cmpt 5193   I cid 5535  β—‘ccnv 5637   β†Ύ cres 5640   ∘ ccom 5642  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362   ↑m cmap 8772  Fincfn 8890  0cc0 11058  1c1 11059  β„•0cn0 12420  β„€cz 12506  β„€β‰₯cuz 12770  ...cfz 13431  mzPolycmzp 41074  Diophcdioph 41107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-hash 14238  df-mzpcl 41075  df-mzp 41076  df-dioph 41108
This theorem is referenced by:  eldioph2b  41115  diophin  41124  diophun  41125
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