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Theorem 2rexfrabdioph 42023
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 42011 . . . 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 3430 . . 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 12509 . . . . . 6 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
53, 4eqeltrid 2829 . . . . 5 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ β„•0)
65adantr 480 . . . 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 42018 . . . . . . . . 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 3829 . . . . . . . 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 5965 . . . . . . . . . 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 42020 . . . . . . . . 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 13541 . . . . . . . . . . . 12 (1...𝑁) βŠ† (1...(𝑁 + 1))
123oveq2i 7412 . . . . . . . . . . . 12 (1...𝑀) = (1...(𝑁 + 1))
1311, 12sseqtrri 4011 . . . . . . . . . . 11 (1...𝑁) βŠ† (1...𝑀)
14 resabs1 6001 . . . . . . . . . . 11 ((1...𝑁) βŠ† (1...𝑀) β†’ ((𝑑 β†Ύ (1...𝑀)) β†Ύ (1...𝑁)) = (𝑑 β†Ύ (1...𝑁)))
15 dfsbcq 3771 . . . . . . . . . . 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 3470 . . . . . . . . . . . . . 14 𝑑 ∈ V
1817resex 6019 . . . . . . . . . . . . 13 (𝑑 β†Ύ (1...𝑀)) ∈ V
19 fveq1 6880 . . . . . . . . . . . . . 14 (π‘Ž = (𝑑 β†Ύ (1...𝑀)) β†’ (π‘Žβ€˜π‘€) = ((𝑑 β†Ύ (1...𝑀))β€˜π‘€))
2019sbcco3gw 4414 . . . . . . . . . . . . 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 12508 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
233, 22eqeltrid 2829 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ β„•)
24 elfz1end 13528 . . . . . . . . . . . . . 14 (𝑀 ∈ β„• ↔ 𝑀 ∈ (1...𝑀))
2523, 24sylib 217 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑀 ∈ (1...𝑀))
26 fvres 6900 . . . . . . . . . . . . 13 (𝑀 ∈ (1...𝑀) β†’ ((𝑑 β†Ύ (1...𝑀))β€˜π‘€) = (π‘‘β€˜π‘€))
27 dfsbcq 3771 . . . . . . . . . . . . 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 283 . . . . . . . . . . 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 3828 . . . . . . . . . 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 283 . . . . . . . . 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 283 . . . . . . . 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 284 . . . . . . 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 3432 . . . . . 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 2810 . . . . 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 476 . . . 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 42022 . . . 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 583 . . 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 2829 . 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 42022 . 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 590 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 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062  {crab 3424  Vcvv 3466  [wsbc 3769   βŠ† wss 3940   β†Ύ cres 5668  β€˜cfv 6533  (class class class)co 7401   ↑m cmap 8816  1c1 11107   + caddc 11109  β„•cn 12209  β„•0cn0 12469  ...cfz 13481  Diophcdioph 41982
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 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  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 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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  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 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-hash 14288  df-mzpcl 41950  df-mzp 41951  df-dioph 41983
This theorem is referenced by:  3rexfrabdioph  42024  4rexfrabdioph  42025  6rexfrabdioph  42026
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