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Theorem 2rexfrabdioph 41305
Description: Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
rexfrabdioph.1 𝑀 = (𝑁 + 1)
rexfrabdioph.2 𝐿 = (𝑀 + 1)
Assertion
Ref Expression
2rexfrabdioph ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} ∈ (Diophβ€˜πΏ)) β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 βˆƒπ‘€ ∈ β„•0 πœ‘} ∈ (Diophβ€˜π‘))
Distinct variable groups:   𝑒,𝑑,𝑣,𝑀,𝐿   𝑑,𝑀,𝑒,𝑣,𝑀   𝑑,𝑁,𝑒,𝑣,𝑀   πœ‘,𝑑
Allowed substitution hints:   πœ‘(𝑀,𝑣,𝑒)
Proof of Theorem 2rexfrabdioph
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 2sbcrex 41293 . . . 4 ([(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]βˆƒπ‘€ ∈ β„•0 πœ‘ ↔ βˆƒπ‘€ ∈ β„•0 [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘)
21rabbii 3437 . . 3 {π‘Ž ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]βˆƒπ‘€ ∈ β„•0 πœ‘} = {π‘Ž ∈ (β„•0 ↑m (1...𝑀)) ∣ βˆƒπ‘€ ∈ β„•0 [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘}
3 rexfrabdioph.1 . . . . . 6 𝑀 = (𝑁 + 1)
4 peano2nn0 12494 . . . . . 6 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
53, 4eqeltrid 2836 . . . . 5 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ β„•0)
65adantr 481 . . . 4 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} ∈ (Diophβ€˜πΏ)) β†’ 𝑀 ∈ β„•0)
7 sbcrot3 41300 . . . . . . . . 9 ([(π‘‘β€˜πΏ) / 𝑀][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘ ↔ [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘)
87sbcbii 3833 . . . . . . . 8 ([(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘‘β€˜πΏ) / 𝑀][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘ ↔ [(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘)
9 reseq1 5967 . . . . . . . . . 10 (π‘Ž = (𝑑 β†Ύ (1...𝑀)) β†’ (π‘Ž β†Ύ (1...𝑁)) = ((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)))
109sbccomieg 41302 . . . . . . . . 9 ([(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) / 𝑒][(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘)
11 fzssp1 13526 . . . . . . . . . . . 12 (1...𝑁) βŠ† (1...(𝑁 + 1))
123oveq2i 7404 . . . . . . . . . . . 12 (1...𝑀) = (1...(𝑁 + 1))
1311, 12sseqtrri 4015 . . . . . . . . . . 11 (1...𝑁) βŠ† (1...𝑀)
14 resabs1 6003 . . . . . . . . . . 11 ((1...𝑁) βŠ† (1...𝑀) β†’ ((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) = (𝑑 β†Ύ (1...𝑁)))
15 dfsbcq 3775 . . . . . . . . . . 11 (((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) = (𝑑 β†Ύ (1...𝑁)) β†’ ([((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) / 𝑒][(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
1613, 14, 15mp2b 10 . . . . . . . . . 10 ([((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) / 𝑒][(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘)
17 vex 3477 . . . . . . . . . . . . . 14 𝑑 ∈ V
1817resex 6021 . . . . . . . . . . . . 13 (𝑑 β†Ύ (1...𝑀)) ∈ V
19 fveq1 6877 . . . . . . . . . . . . . 14 (π‘Ž = (𝑑 β†Ύ (1...𝑀)) β†’ (π‘Žβ€˜π‘€) = ((𝑑 β†Ύ (1...𝑀))β€˜π‘€))
2019sbcco3gw 4418 . . . . . . . . . . . . 13 ((𝑑 β†Ύ (1...𝑀)) ∈ V β†’ ([(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [((𝑑 β†Ύ (1...𝑀))β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
2118, 20ax-mp 5 . . . . . . . . . . . 12 ([(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [((𝑑 β†Ύ (1...𝑀))β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘)
22 nn0p1nn 12493 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
233, 22eqeltrid 2836 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ β„•)
24 elfz1end 13513 . . . . . . . . . . . . . 14 (𝑀 ∈ β„• ↔ 𝑀 ∈ (1...𝑀))
2523, 24sylib 217 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ (1...𝑀))
26 fvres 6897 . . . . . . . . . . . . 13 (𝑀 ∈ (1...𝑀) β†’ ((𝑑 β†Ύ (1...𝑀))β€˜π‘€) = (π‘‘β€˜π‘€))
27 dfsbcq 3775 . . . . . . . . . . . . 13 (((𝑑 β†Ύ (1...𝑀))β€˜π‘€) = (π‘‘β€˜π‘€) β†’ ([((𝑑 β†Ύ (1...𝑀))β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
2825, 26, 273syl 18 . . . . . . . . . . . 12 (𝑁 ∈ β„•0 β†’ ([((𝑑 β†Ύ (1...𝑀))β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
2921, 28bitrid 282 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ ([(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
3029sbcbidv 3832 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ ([(𝑑 β†Ύ (1...𝑁)) / 𝑒][(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
3116, 30bitrid 282 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ ([((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) / 𝑒][(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
3210, 31bitrid 282 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ ([(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘))
338, 32bitr2id 283 . . . . . . 7 (𝑁 ∈ β„•0 β†’ ([(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘ ↔ [(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘‘β€˜πΏ) / 𝑀][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘))
3433rabbidv 3439 . . . . . 6 (𝑁 ∈ β„•0 β†’ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} = {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘‘β€˜πΏ) / 𝑀][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘})
3534eleq1d 2817 . . . . 5 (𝑁 ∈ β„•0 β†’ ({𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} ∈ (Diophβ€˜πΏ) ↔ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘‘β€˜πΏ) / 𝑀][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘} ∈ (Diophβ€˜πΏ)))
3635biimpa 477 . . . 4 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} ∈ (Diophβ€˜πΏ)) β†’ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘‘β€˜πΏ) / 𝑀][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘} ∈ (Diophβ€˜πΏ))
37 rexfrabdioph.2 . . . . 5 𝐿 = (𝑀 + 1)
3837rexfrabdioph 41304 . . . 4 ((𝑀 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑀)) / π‘Ž][(π‘‘β€˜πΏ) / 𝑀][(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘} ∈ (Diophβ€˜πΏ)) β†’ {π‘Ž ∈ (β„•0 ↑m (1...𝑀)) ∣ βˆƒπ‘€ ∈ β„•0 [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘} ∈ (Diophβ€˜π‘€))
396, 36, 38syl2anc 584 . . 3 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} ∈ (Diophβ€˜πΏ)) β†’ {π‘Ž ∈ (β„•0 ↑m (1...𝑀)) ∣ βˆƒπ‘€ ∈ β„•0 [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]πœ‘} ∈ (Diophβ€˜π‘€))
402, 39eqeltrid 2836 . 2 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} ∈ (Diophβ€˜πΏ)) β†’ {π‘Ž ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]βˆƒπ‘€ ∈ β„•0 πœ‘} ∈ (Diophβ€˜π‘€))
413rexfrabdioph 41304 . 2 ((𝑁 ∈ β„•0 ∧ {π‘Ž ∈ (β„•0 ↑m (1...𝑀)) ∣ [(π‘Ž β†Ύ (1...𝑁)) / 𝑒][(π‘Žβ€˜π‘€) / 𝑣]βˆƒπ‘€ ∈ β„•0 πœ‘} ∈ (Diophβ€˜π‘€)) β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 βˆƒπ‘€ ∈ β„•0 πœ‘} ∈ (Diophβ€˜π‘))
4240, 41syldan 591 1 ((𝑁 ∈ β„•0 ∧ {𝑑 ∈ (β„•0 ↑m (1...𝐿)) ∣ [(𝑑 β†Ύ (1...𝑁)) / 𝑒][(π‘‘β€˜π‘€) / 𝑣][(π‘‘β€˜πΏ) / 𝑀]πœ‘} ∈ (Diophβ€˜πΏ)) β†’ {𝑒 ∈ (β„•0 ↑m (1...𝑁)) ∣ βˆƒπ‘£ ∈ β„•0 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  βˆƒwrex 3069  {crab 3431  Vcvv 3473  [wsbc 3773   βŠ† wss 3944   β†Ύ cres 5671  β€˜cfv 6532  (class class class)co 7393   ↑m cmap 8803  1c1 11093   + caddc 11095  β„•cn 12194  β„•0cn0 12454  ...cfz 13466  Diophcdioph 41264
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-oadd 8452  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-dju 9878  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-hash 14273  df-mzpcl 41232  df-mzp 41233  df-dioph 41265
This theorem is referenced by:  3rexfrabdioph  41306  4rexfrabdioph  41307  6rexfrabdioph  41308
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