ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genppreclu GIF version

Theorem genppreclu 7330
Description: Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genppreclu ((𝐴P𝐵P) → ((𝐶 ∈ (2nd𝐴) ∧ 𝐷 ∈ (2nd𝐵)) → (𝐶𝐺𝐷) ∈ (2nd ‘(𝐴𝐹𝐵))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑤,𝑣
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐷(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genppreclu
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2139 . . 3 (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2 rspceov 5813 . . 3 ((𝐶 ∈ (2nd𝐴) ∧ 𝐷 ∈ (2nd𝐵) ∧ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)) → ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)(𝐶𝐺𝐷) = (𝑔𝐺))
31, 2mp3an3 1304 . 2 ((𝐶 ∈ (2nd𝐴) ∧ 𝐷 ∈ (2nd𝐵)) → ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)(𝐶𝐺𝐷) = (𝑔𝐺))
4 genpelvl.1 . . 3 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
5 genpelvl.2 . . 3 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
64, 5genpelvu 7328 . 2 ((𝐴P𝐵P) → ((𝐶𝐺𝐷) ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)(𝐶𝐺𝐷) = (𝑔𝐺)))
73, 6syl5ibr 155 1 ((𝐴P𝐵P) → ((𝐶 ∈ (2nd𝐴) ∧ 𝐷 ∈ (2nd𝐵)) → (𝐶𝐺𝐷) ∈ (2nd ‘(𝐴𝐹𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wcel 1480  wrex 2417  {crab 2420  cop 3530  cfv 5123  (class class class)co 5774  cmpo 5776  1st c1st 6036  2nd c2nd 6037  Qcnq 7095  Pcnp 7106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7119  df-nqqs 7163  df-inp 7281
This theorem is referenced by:  genpmu  7333  genprndu  7337  addnqpru  7345  mulnqpru  7384  distrlem1pru  7398  distrlem4pru  7400  ltexprlemru  7427  addcanprleml  7429  addcanprlemu  7430
  Copyright terms: Public domain W3C validator