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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexrabdioph Structured version   Visualization version   GIF version
Theorem rexrabdioph 41517
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 3433 . . . . . 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 3778 . . . . . . . . . . 11 (𝑏 = 𝑐 β†’ ([𝑏 / 𝑣][π‘Ž / 𝑒]πœ“ ↔ [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“))
32cbvrexvw 3235 . . . . . . . . . 10 (βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“ ↔ βˆƒπ‘ ∈ β„•0 [𝑐 / 𝑣][π‘Ž / 𝑒]πœ“)
43anbi2i 623 . . . . . . . . 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 3190 . . . . . . . . 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 277 . . . . . . . 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 765 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ 𝑁 ∈ β„•0)
8 simpr 485 . . . . . . . . . . . . 13 (((𝑁 ∈ β„•0 ∧ 𝑐 ∈ β„•0) ∧ π‘Ž ∈ (β„•0 ↑m (1...𝑁))) β†’ π‘Ž ∈ (β„•0 ↑m (1...𝑁)))
9 simplr 767 . . . . . . . . . . . . 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 41439 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 715 . . . . . . . . . . 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 41442 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . 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 41440 . . . . . . . . . . . . . . . . 17 (π‘Ž ∈ (β„•0 ↑m (1...𝑁)) β†’ ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†Ύ (1...𝑁)) = π‘Ž)
1817adantl 482 . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 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 3781 . . . . . . . . . . . . . 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 3783 . . . . . . . . . . . . 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 455 . . . . . . . . . . 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 715 . . . . . . . . . . 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 6887 . . . . . . . . . . . . . 14 (𝑏 = (π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©}) β†’ (π‘β€˜π‘€) = ((π‘Ž βˆͺ {βŸ¨π‘€, π‘βŸ©})β€˜π‘€))
26 reseq1 5973 . . . . . . . . . . . . . . 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 3781 . . . . . . . . . . . . . 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 3783 . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . 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 631 . . . . . . . . . . . 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 3612 . . . . . . . . . . 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 835 . . . . . . . . . 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 3157 . . . . . . . . 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 8839 . . . . . . . . . . . . 13 (𝑏 ∈ (β„•0 ↑m (1...𝑀)) β†’ 𝑏:(1...𝑀)βŸΆβ„•0)
35 nn0p1nn 12507 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
3610, 35eqeltrid 2837 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ β„•)
37 elfz1end 13527 . . . . . . . . . . . . . 14 (𝑀 ∈ β„• ↔ 𝑀 ∈ (1...𝑀))
3836, 37sylib 217 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ (1...𝑀))
39 ffvelcdm 7080 . . . . . . . . . . . . 13 ((𝑏:(1...𝑀)βŸΆβ„•0 ∧ 𝑀 ∈ (1...𝑀)) β†’ (π‘β€˜π‘€) ∈ β„•0)
4034, 38, 39syl2anr 597 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑏 ∈ (β„•0 ↑m (1...𝑀))) β†’ (π‘β€˜π‘€) ∈ β„•0)
4140adantr 481 . . . . . . . . . . 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 771 . . . . . . . . . . . 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 41441 . . . . . . . . . . . . 13 (𝑏 ∈ (β„•0 ↑m (1...𝑀)) β†’ (𝑏 β†Ύ (1...𝑁)) ∈ (β„•0 ↑m (1...𝑁)))
4443ad2antlr 725 . . . . . . . . . . . 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 2833 . . . . . . . . . . 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 769 . . . . . . . . . . . 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 3778 . . . . . . . . . . . . . 14 (π‘Ž = (𝑏 β†Ύ (1...𝑁)) β†’ ([π‘Ž / 𝑒]πœ“ ↔ [(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“))
4847sbcbidv 3835 . . . . . . . . . . . . 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 727 . . . . . . . . . . . 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 256 . . . . . . . . . . 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 3778 . . . . . . . . . . . . 13 (𝑐 = (π‘β€˜π‘€) β†’ ([𝑐 / 𝑣][π‘Ž / 𝑒]πœ“ ↔ [(π‘β€˜π‘€) / 𝑣][π‘Ž / 𝑒]πœ“))
5251anbi2d 629 . . . . . . . . . . . 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 3612 . . . . . . . . . . 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 835 . . . . . . . . . 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 3157 . . . . . . . . 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 282 . . . . . . 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 2801 . . . . . 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 2784 . . . . 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 2903 . . . . . 6 Ⅎ𝑒(β„•0 ↑m (1...𝑁))
61 nfcv 2903 . . . . . 6 β„²π‘Ž(β„•0 ↑m (1...𝑁))
62 nfv 1917 . . . . . 6 β„²π‘Žβˆƒπ‘£ ∈ β„•0 πœ“
63 nfcv 2903 . . . . . . 7 Ⅎ𝑒ℕ0
64 nfcv 2903 . . . . . . . 8 Ⅎ𝑒𝑏
65 nfsbc1v 3796 . . . . . . . 8 Ⅎ𝑒[π‘Ž / 𝑒]πœ“
6664, 65nfsbcw 3798 . . . . . . 7 Ⅎ𝑒[𝑏 / 𝑣][π‘Ž / 𝑒]πœ“
6763, 66nfrexw 3310 . . . . . 6 β„²π‘’βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“
68 sbceq1a 3787 . . . . . . . 8 (𝑒 = π‘Ž β†’ (πœ“ ↔ [π‘Ž / 𝑒]πœ“))
6968rexbidv 3178 . . . . . . 7 (𝑒 = π‘Ž β†’ (βˆƒπ‘£ ∈ β„•0 πœ“ ↔ βˆƒπ‘£ ∈ β„•0 [π‘Ž / 𝑒]πœ“))
70 nfv 1917 . . . . . . . 8 Ⅎ𝑏[π‘Ž / 𝑒]πœ“
71 nfsbc1v 3796 . . . . . . . 8 Ⅎ𝑣[𝑏 / 𝑣][π‘Ž / 𝑒]πœ“
72 sbceq1a 3787 . . . . . . . 8 (𝑣 = 𝑏 β†’ ([π‘Ž / 𝑒]πœ“ ↔ [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“))
7370, 71, 72cbvrexw 3304 . . . . . . 7 (βˆƒπ‘£ ∈ β„•0 [π‘Ž / 𝑒]πœ“ ↔ βˆƒπ‘ ∈ β„•0 [𝑏 / 𝑣][π‘Ž / 𝑒]πœ“)
7469, 73bitrdi 286 . . . . . 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 3467 . . . . 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 6887 . . . . . . . 8 (𝑑 = 𝑏 β†’ (π‘‘β€˜π‘€) = (π‘β€˜π‘€))
77 reseq1 5973 . . . . . . . . 9 (𝑑 = 𝑏 β†’ (𝑑 β†Ύ (1...𝑁)) = (𝑏 β†Ύ (1...𝑁)))
7877sbceq1d 3781 . . . . . . . 8 (𝑑 = 𝑏 β†’ ([(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“ ↔ [(𝑏 β†Ύ (1...𝑁)) / 𝑒]πœ“))
7976, 78sbceqbid 3783 . . . . . . 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 3691 . . . . . 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 2802 . . . . 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 2797 . . . 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 6901 . . . . . . . 8 (π‘‘β€˜π‘€) ∈ V
84 vex 3478 . . . . . . . . 9 𝑑 ∈ V
8584resex 6027 . . . . . . . 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 510 . . . . . . . 8 ((𝑣 = (π‘‘β€˜π‘€) ∧ 𝑒 = (𝑑 β†Ύ (1...𝑁))) β†’ (πœ“ ↔ πœ‘))
8983, 85, 88sbc2ie 3859 . . . . . . 7 ([(π‘‘β€˜π‘€) / 𝑣][(𝑑 β†Ύ (1...𝑁)) / 𝑒]πœ“ ↔ πœ‘)
9089rabbii 3438 . . . . . 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 3324 . . . . 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 2802 . . . 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 2788 . . 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 481 . 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 483 . . 3 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ 𝑁 ∈ β„•0)
96 nn0z 12579 . . . . . 6 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„€)
97 uzid 12833 . . . . . 6 (𝑁 ∈ β„€ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘))
98 peano2uz 12881 . . . . . 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 2837 . . . 4 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘))
101100adantr 481 . . 3 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝑀)) ∣ πœ‘} ∈ (Diophβ€˜π‘€)) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘))
102 simpr 485 . . 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 41498 . . 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 1371 . 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 2833 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆƒwrex 3070  {crab 3432  [wsbc 3776   βˆͺ cun 3945  {csn 4627  βŸ¨cop 4633   β†Ύ cres 5677  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  1c1 11107   + caddc 11109  β„•cn 12208  β„•0cn0 12468  β„€cz 12554  β„€β‰₯cuz 12818  ...cfz 13480  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-inf2 9632  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:  rexfrabdioph  41518  elnn0rabdioph  41526  dvdsrabdioph  41533
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