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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexrabdioph Structured version   Visualization version   GIF version
Theorem rexrabdioph 41532
Description: Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Hypotheses
Ref Expression
rexrabdioph.1 𝑀 = (𝑁 + 1)
rexrabdioph.2 (𝑣 = (π‘‘β€˜π‘€) β†’ (πœ“ ↔ πœ’))
rexrabdioph.3 (𝑒 = (𝑑 β†Ύ (1...𝑁)) β†’ (πœ’ ↔ πœ‘))
Assertion
Ref Expression
rexrabdioph ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 πœ“} ∈ (Diophβ€˜π‘))
Distinct variable groups:   𝑑,𝑁,𝑒,𝑣   𝑑,𝑀,𝑒,𝑣   πœ‘,𝑒,𝑣   πœ“,𝑑   πœ’,𝑣
Allowed substitution hints:   πœ‘(𝑑)   πœ“(𝑣,𝑒)   πœ’(𝑒,𝑑)
Proof of Theorem rexrabdioph
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 3434 . . . . . 6 {π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“} = {π‘Ž ∣ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“)}
2 dfsbcq 3780 . . . . . . . . . . 11 (𝑏 = 𝑐 β†’ ([𝑏 / 𝑣][π‘Ž / 𝑒]πœ“ ↔ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“))
32cbvrexvw 3236 . . . . . . . . . 10 (βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“ ↔ βˆƒπ‘ ∈ β„•0 [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“)
43anbi2i 624 . . . . . . . . 9 ((π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“) ↔ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘ ∈ β„•0 [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“))
5 r19.42v 3191 . . . . . . . . 9 (βˆƒπ‘ ∈ β„•0 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“) ↔ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘ ∈ β„•0 [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“))
64, 5bitr4i 278 . . . . . . . 8 ((π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“) ↔ βˆƒπ‘ ∈ β„•0 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“))
7 simpll 766 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ 𝑁 ∈ β„•0)
8 simpr 486 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ π‘Ž ∈ (β„•0 ↑m (1...𝑁)))
9 simplr 768 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ 𝑐 ∈ β„•0)
10 rexrabdioph.1 . . . . . . . . . . . . . 14 𝑀 = (𝑁 + 1)
1110mapfzcons 41454 . . . . . . . . . . . . 13 ((𝑁 ∈ β„•0 ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑐 ∈ β„•0) β†’ (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) ∈ (β„•0 ↑m (1...𝑀)))
127, 8, 9, 11syl3anc 1372 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) ∈ (β„•0 ↑m (1...𝑀)))
1312adantrr 716 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“)) β†’ (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) ∈ (β„•0 ↑m (1...𝑀)))
1410mapfzcons2 41457 . . . . . . . . . . . . . . . 16 ((π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ 𝑐 ∈ β„•0) β†’ ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) = 𝑐)
158, 9, 14syl2anc 585 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) = 𝑐)
1615eqcomd 2739 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ 𝑐 = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€))
1710mapfzcons1 41455 . . . . . . . . . . . . . . . . 17 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) β†’ ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) = π‘Ž)
1817adantl 483 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) = π‘Ž)
1918eqcomd 2739 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ π‘Ž = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)))
2019sbceq1d 3783 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ ([π‘Ž / 𝑒]πœ“ ↔ [((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“))
2116, 20sbceqbid 3785 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ ([𝑐 / 𝑣][π‘Ž / 𝑒]πœ“ ↔ [((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) / 𝑣][((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“))
2221biimpd 228 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ ([𝑐 / 𝑣][π‘Ž / 𝑒]πœ“ β†’ [((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) / 𝑣][((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“))
2322impr 456 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“)) β†’ [((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) / 𝑣][((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“)
2419adantrr 716 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“)) β†’ π‘Ž = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)))
25 fveq1 6891 . . . . . . . . . . . . . 14 (𝑏 = (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†’ (π‘β€˜π‘€) = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€))
26 reseq1 5976 . . . . . . . . . . . . . . 15 (𝑏 = (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†’ (𝑏 β†Ύ (1...𝑁)) = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)))
2726sbceq1d 3783 . . . . . . . . . . . . . 14 (𝑏 = (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†’ ([(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ↔ [((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“))
2825, 27sbceqbid 3785 . . . . . . . . . . . . 13 (𝑏 = (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†’ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ↔ [((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) / 𝑣][((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“))
2926eqeq2d 2744 . . . . . . . . . . . . 13 (𝑏 = (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†’ (π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ π‘Ž = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁))))
3028, 29anbi12d 632 . . . . . . . . . . . 12 (𝑏 = (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†’ (([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) ↔ ([((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) / 𝑣][((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)))))
3130rspcev 3613 . . . . . . . . . . 11 (((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) ∈ (β„•0 ↑m (1...𝑀)) ∧ ([((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€) / 𝑣][((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)))) β†’ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))))
3213, 23, 24, 31syl12anc 836 . . . . . . . . . 10 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“)) β†’ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))))
3332rexlimdva2 3158 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (βˆƒπ‘ ∈ β„•0 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“) β†’ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))))
34 elmapi 8843 . . . . . . . . . . . . 13 (𝑏 ∈ (β„•0 ↑m (1...𝑀)) β†’ 𝑏:(1...𝑀)βŸΆβ„•0)
35 nn0p1nn 12511 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
3610, 35eqeltrid 2838 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ β„•)
37 elfz1end 13531 . . . . . . . . . . . . . 14 (𝑀 ∈ β„• ↔ 𝑀 ∈ (1...𝑀))
3836, 37sylib 217 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ (1...𝑀))
39 ffvelcdm 7084 . . . . . . . . . . . . 13 ((𝑏:(1...𝑀)βŸΆβ„•0 ∧ 𝑀 ∈ (1...𝑀)) β†’ (π‘β€˜π‘€) ∈ β„•0)
4034, 38, 39syl2anr 598 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) β†’ (π‘β€˜π‘€) ∈ β„•0)
4140adantr 482 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ (π‘β€˜π‘€) ∈ β„•0)
42 simprr 772 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ π‘Ž = (𝑏 β†Ύ (1...𝑁)))
4310mapfzcons1cl 41456 . . . . . . . . . . . . 13 (𝑏 ∈ (β„•0 ↑m (1...𝑀)) β†’ (𝑏 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)))
4443ad2antlr 726 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ (𝑏 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)))
4542, 44eqeltrd 2834 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ π‘Ž ∈ (β„•0 ↑m (1...𝑁)))
46 simprl 770 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ [(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“)
47 dfsbcq 3780 . . . . . . . . . . . . . 14 (π‘Ž = (𝑏 β†Ύ (1...𝑁)) β†’ ([π‘Ž / 𝑒]πœ“ ↔ [(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“))
4847sbcbidv 3837 . . . . . . . . . . . . 13 (π‘Ž = (𝑏 β†Ύ (1...𝑁)) β†’ ([(π‘β€˜π‘€) / 𝑣][π‘Ž / 𝑒]πœ“ ↔ [(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“))
4948ad2antll 728 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ ([(π‘β€˜π‘€) / 𝑣][π‘Ž / 𝑒]πœ“ ↔ [(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“))
5046, 49mpbird 257 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ [(π‘β€˜π‘€) / 𝑣][π‘Ž / 𝑒]πœ“)
51 dfsbcq 3780 . . . . . . . . . . . . 13 (𝑐 = (π‘β€˜π‘€) β†’ ([𝑐 / 𝑣][π‘Ž / 𝑒]πœ“ ↔ [(π‘β€˜π‘€) / 𝑣][π‘Ž / 𝑒]πœ“))
5251anbi2d 630 . . . . . . . . . . . 12 (𝑐 = (π‘β€˜π‘€) β†’ ((π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“) ↔ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [(π‘β€˜π‘€) / 𝑣][π‘Ž / 𝑒]πœ“)))
5352rspcev 3613 . . . . . . . . . . 11 (((π‘β€˜π‘€) ∈ β„•0 ∧ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [(π‘β€˜π‘€) / 𝑣][π‘Ž / 𝑒]πœ“)) β†’ βˆƒπ‘ ∈ β„•0 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“))
5441, 45, 50, 53syl12anc 836 . . . . . . . . . 10 (((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) ∧ ([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))) β†’ βˆƒπ‘ ∈ β„•0 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“))
5554rexlimdva2 3158 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))) β†’ βˆƒπ‘ ∈ β„•0 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“)))
5633, 55impbid 211 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ (βˆƒπ‘ ∈ β„•0 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“) ↔ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))))
576, 56bitrid 283 . . . . . . 7 (𝑁 ∈ β„•0 β†’ ((π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“) ↔ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))))
5857abbidv 2802 . . . . . 6 (𝑁 ∈ β„•0 β†’ {π‘Ž ∣ (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∧ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“)} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))})
591, 58eqtrid 2785 . . . . 5 (𝑁 ∈ β„•0 β†’ {π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))})
60 nfcv 2904 . . . . . 6 Ⅎ𝑒(β„•0 ↑m (1...𝑁))
61 nfcv 2904 . . . . . 6 β„²π‘Ž(β„•0 ↑m (1...𝑁))
62 nfv 1918 . . . . . 6 β„²π‘Žβˆƒπ‘£ ∈ β„•0 πœ“
63 nfcv 2904 . . . . . . 7 Ⅎ𝑒ℕ0
64 nfcv 2904 . . . . . . . 8 Ⅎ𝑒𝑏
65 nfsbc1v 3798 . . . . . . . 8 Ⅎ𝑒[π‘Ž / 𝑒]πœ“
6664, 65nfsbcw 3800 . . . . . . 7 Ⅎ𝑒[𝑏 / 𝑣][π‘Ž / 𝑒]πœ“
6763, 66nfrexw 3311 . . . . . 6 β„²π‘’βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“
68 sbceq1a 3789 . . . . . . . 8 (𝑒 = π‘Ž β†’ (πœ“ ↔ [π‘Ž / 𝑒]πœ“))
6968rexbidv 3179 . . . . . . 7 (𝑒 = π‘Ž β†’ (βˆƒπ‘£ ∈ β„•0 πœ“ ↔ βˆƒπ‘£ ∈ β„•0 [π‘Ž / 𝑒]πœ“))
70 nfv 1918 . . . . . . . 8 Ⅎ𝑏[π‘Ž / 𝑒]πœ“
71 nfsbc1v 3798 . . . . . . . 8 Ⅎ𝑣[𝑏 / 𝑣][π‘Ž / 𝑒]πœ“
72 sbceq1a 3789 . . . . . . . 8 (𝑣 = 𝑏 β†’ ([π‘Ž / 𝑒]πœ“ ↔ [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“))
7370, 71, 72cbvrexw 3305 . . . . . . 7 (βˆƒπ‘£ ∈ β„•0 [π‘Ž / 𝑒]πœ“ ↔ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“)
7469, 73bitrdi 287 . . . . . 6 (𝑒 = π‘Ž β†’ (βˆƒπ‘£ ∈ β„•0 πœ“ ↔ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“))
7560, 61, 62, 67, 74cbvrabw 3468 . . . . 5 {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 πœ“} = {π‘Ž ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“}
76 fveq1 6891 . . . . . . . 8 (𝑑 = 𝑏 β†’ (π‘‘β€˜π‘€) = (π‘β€˜π‘€))
77 reseq1 5976 . . . . . . . . 9 (𝑑 = 𝑏 β†’ (𝑑 β†Ύ (1...𝑁)) = (𝑏 β†Ύ (1...𝑁)))
7877sbceq1d 3783 . . . . . . . 8 (𝑑 = 𝑏 β†’ ([(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“ ↔ [(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“))
7976, 78sbceqbid 3785 . . . . . . 7 (𝑑 = 𝑏 β†’ ([(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“ ↔ [(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“))
8079rexrab 3693 . . . . . 6 (βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“}π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁))))
8180abbii 2803 . . . . 5 {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“}π‘Ž = (𝑏 β†Ύ (1...𝑁))} = {π‘Ž ∣ βˆƒπ‘ ∈ (β„•0 ↑m (1...𝑀))([(π‘β€˜π‘€) / 𝑣][(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“ ∧ π‘Ž = (𝑏 β†Ύ (1...𝑁)))}
8259, 75, 813eqtr4g 2798 . . . 4 (𝑁 ∈ β„•0 β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 πœ“} = {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“}π‘Ž = (𝑏 β†Ύ (1...𝑁))})
83 fvex 6905 . . . . . . . 8 (π‘‘β€˜π‘€) ∈ V
84 vex 3479 . . . . . . . . 9 𝑑 ∈ V
8584resex 6030 . . . . . . . 8 (𝑑 β†Ύ (1...𝑁)) ∈ V
86 rexrabdioph.2 . . . . . . . . 9 (𝑣 = (π‘‘β€˜π‘€) β†’ (πœ“ ↔ πœ’))
87 rexrabdioph.3 . . . . . . . . 9 (𝑒 = (𝑑 β†Ύ (1...𝑁)) β†’ (πœ’ ↔ πœ‘))
8886, 87sylan9bb 511 . . . . . . . 8 ((𝑣 = (π‘‘β€˜π‘€) ∧ 𝑒 = (𝑑 β†Ύ (1...𝑁))) β†’ (πœ“ ↔ πœ‘))
8983, 85, 88sbc2ie 3861 . . . . . . 7 ([(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“ ↔ πœ‘)
9089rabbii 3439 . . . . . 6 {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“} = {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘}
9190rexeqi 3325 . . . . 5 (βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“}π‘Ž = (𝑏 β†Ύ (1...𝑁)) ↔ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘}π‘Ž = (𝑏 β†Ύ (1...𝑁)))
9291abbii 2803 . . . 4 {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“}π‘Ž = (𝑏 β†Ύ (1...𝑁))} = {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘}π‘Ž = (𝑏 β†Ύ (1...𝑁))}
9382, 92eqtrdi 2789 . . 3 (𝑁 ∈ β„•0 β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 πœ“} = {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘}π‘Ž = (𝑏 β†Ύ (1...𝑁))})
9493adantr 482 . 2 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 πœ“} = {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘}π‘Ž = (𝑏 β†Ύ (1...𝑁))})
95 simpl 484 . . 3 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ 𝑁 ∈ β„•0)
96 nn0z 12583 . . . . . 6 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„€)
97 uzid 12837 . . . . . 6 (𝑁 ∈ β„€ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘))
98 peano2uz 12885 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜π‘) β†’ (𝑁 + 1) ∈ (β„€β‰₯β€˜π‘))
9996, 97, 983syl 18 . . . . 5 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ (β„€β‰₯β€˜π‘))
10010, 99eqeltrid 2838 . . . 4 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘))
101100adantr 482 . . 3 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘))
102 simpr 486 . . 3 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€))
103 diophrex 41513 . . 3 ((𝑁 ∈ β„•0 ∧ 𝑀 ∈ (β„€β‰₯β€˜π‘) ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘}π‘Ž = (𝑏 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
10495, 101, 102, 103syl3anc 1372 . 2 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ {π‘Ž ∣ βˆƒπ‘ ∈ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘}π‘Ž = (𝑏 β†Ύ (1...𝑁))} ∈ (Diophβ€˜π‘))
10594, 104eqeltrd 2834 1 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 πœ“} ∈ (Diophβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071  {crab 3433  [wsbc 3778   βˆͺ cun 3947  {csn 4629  βŸ¨cop 4635   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  1c1 11111   + caddc 11113  β„•cn 12212  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  ...cfz 13484  Diophcdioph 41493
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-mzpcl 41461  df-mzp 41462  df-dioph 41494
This theorem is referenced by:  rexfrabdioph  41533  elnn0rabdioph  41541  dvdsrabdioph  41548
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