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Theorem eldioph2 42078
Description: Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 42068. (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 42068 . . 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 1132 . 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 6884 . . . . . . . . . 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 2728 . . . . . . . . 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 628 . . . . . . . 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 3172 . . . . . . 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 2795 . . . . . 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 726 . . . . 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 772 . . . . . . . . . . 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 774 . . . . . . . . . . . 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 13943 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
12 unfi 9174 . . . . . . . . . . . 12 ((π‘Ž ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ (π‘Ž βˆͺ (1...𝑁)) ∈ Fin)
1310, 11, 12sylancl 585 . . . . . . . . . . 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 4168 . . . . . . . . . . . 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 42076 . . . . . . . . . . 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 1368 . . . . . . . . . 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 6854 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ (𝑑 ∘ ◑𝑑) = ( I β†Ύ (π‘Ž βˆͺ (1...𝑁))))
1918ad2antrl 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4167 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 π‘Ž βŠ† (π‘Ž βˆͺ (1...𝑁))
22 resabs1 6005 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 ∘ ◑𝑑) β†Ύ π‘Ž) = (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž))
2624, 25eqtrdi 2782 . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . 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 5856 . . . . . . . . . . . . . . . . . . . . . . 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 6257 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ ( I β†Ύ π‘Ž)) = (𝑒 β†Ύ π‘Ž)
30 coass 6258 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž)) = (𝑒 ∘ (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž)))
3130eqcomi 2735 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 ∘ (𝑑 ∘ (◑𝑑 β†Ύ π‘Ž))) = ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))
3228, 29, 313eqtr3g 2789 . . . . . . . . . . . . . . . . . . . . . 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 6889 . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . 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 6826 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ 𝑑:(1...𝑐)–1-1β†’(π‘Ž βˆͺ (1...𝑁)))
3736ad2antrl 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†’ 𝑑:(1...𝑐)–1-1β†’(π‘Ž βˆͺ (1...𝑁)))
38 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) β†’ (1...𝑁) βŠ† 𝑆)
4039ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4181 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6787 . . . . . . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . . . . . . 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 6781 . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . 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 42030 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑐) ∈ V ∧ 𝑒 ∈ (β„€ ↑m 𝑆) ∧ 𝑑:(1...𝑐)βŸΆπ‘†) β†’ (𝑒 ∘ 𝑑) ∈ (β„€ ↑m (1...𝑐)))
4934, 35, 47, 48syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 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 5851 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„Ž = (𝑒 ∘ 𝑑) β†’ (β„Ž ∘ (◑𝑑 β†Ύ π‘Ž)) = ((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž)))
5150fveq2d 6889 . . . . . . . . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . . . . . . . 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 6898 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘β€˜((𝑒 ∘ 𝑑) ∘ (◑𝑑 β†Ύ π‘Ž))) ∈ V
5451, 52, 53fvmpt 6992 . . . . . . . . . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . . . . . . . . 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 5241 . . . . . . . . . . . . . . . . . . 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 6887 . . . . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ (𝑆 ∈ V ∧ (1...𝑁) βŠ† 𝑆)) ∧ (π‘Ž ∈ Fin ∧ 𝑏 ∈ (mzPolyβ€˜π‘Ž))) β†’ 𝑆 ∈ V)
6463ad3antrrr 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†’ 𝑆 ∈ V)
65 simprr 770 . . . . . . . . . . . . . . 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 42075 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . 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 6839 . . . . . . . . . . . . . . . . . 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 6827 . . . . . . . . . . . . . . . . . 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 6751 . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . 16 (𝑑:(1...𝑐)–1-1-ontoβ†’(π‘Ž βˆͺ (1...𝑁)) β†’ (◑𝑑 β†Ύ π‘Ž):π‘ŽβŸΆ(1...𝑐))
7978ad2antrl 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†’ (◑𝑑 β†Ύ π‘Ž):π‘ŽβŸΆ(1...𝑐))
80 mzprename 42065 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 42074 . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . 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 2827 . . . . . . . . . . . 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 412 . . . . . . . . . . 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 3205 . . . . . . . . . 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 419 . . . . . . . 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 1129 . . . . . . 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 417 . . . . . 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 714 . . . . 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 2827 . . . 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 412 . . 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 3205 . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943   ↦ cmpt 5224   I cid 5566  β—‘ccnv 5668   β†Ύ cres 5671   ∘ ccom 5673  βŸΆwf 6533  β€“1-1β†’wf1 6534  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405   ↑m cmap 8822  Fincfn 8941  0cc0 11112  1c1 11113  β„•0cn0 12476  β„€cz 12562  β„€β‰₯cuz 12826  ...cfz 13490  mzPolycmzp 42038  Diophcdioph 42071
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-hash 14296  df-mzpcl 42039  df-mzp 42040  df-dioph 42072
This theorem is referenced by:  eldioph2b  42079  diophin  42088  diophun  42089
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