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

Theorem genpdf 7323
Description: Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
Hypothesis
Ref Expression
genpdf.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)
Assertion
Ref Expression
genpdf 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)
Distinct variable group:   𝑟,𝑞,𝑠,𝑣,𝑤
Allowed substitution hints:   𝐹(𝑤,𝑣,𝑠,𝑟,𝑞)   𝐺(𝑤,𝑣,𝑠,𝑟,𝑞)

Proof of Theorem genpdf
StepHypRef Expression
1 genpdf.1 . 2 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)
2 prop 7290 . . . . . . 7 (𝑤P → ⟨(1st𝑤), (2nd𝑤)⟩ ∈ P)
3 elprnql 7296 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (1st𝑤)) → 𝑟Q)
42, 3sylan 281 . . . . . 6 ((𝑤P𝑟 ∈ (1st𝑤)) → 𝑟Q)
54adantlr 468 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (1st𝑤)) → 𝑟Q)
6 prop 7290 . . . . . . 7 (𝑣P → ⟨(1st𝑣), (2nd𝑣)⟩ ∈ P)
7 elprnql 7296 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (1st𝑣)) → 𝑠Q)
86, 7sylan 281 . . . . . 6 ((𝑣P𝑠 ∈ (1st𝑣)) → 𝑠Q)
98adantll 467 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (1st𝑣)) → 𝑠Q)
105, 9genpdflem 7322 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)})
11 elprnqu 7297 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
122, 11sylan 281 . . . . . 6 ((𝑤P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
1312adantlr 468 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (2nd𝑤)) → 𝑟Q)
14 elprnqu 7297 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
156, 14sylan 281 . . . . . 6 ((𝑣P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1615adantll 467 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1713, 16genpdflem 7322 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)})
1810, 17opeq12d 3713 . . 3 ((𝑤P𝑣P) → ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩ = ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)
1918mpoeq3ia 5836 . 2 (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩) = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)
201, 19eqtri 2160 1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)
Colors of variables: wff set class
Syntax hints:  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-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  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-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-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:  genipv  7324
  Copyright terms: Public domain W3C validator