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Theorem genpdf 6663
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 6630 . . . . . . 7 (𝑤P → ⟨(1st𝑤), (2nd𝑤)⟩ ∈ P)
3 elprnql 6636 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (1st𝑤)) → 𝑟Q)
42, 3sylan 271 . . . . . 6 ((𝑤P𝑟 ∈ (1st𝑤)) → 𝑟Q)
54adantlr 454 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (1st𝑤)) → 𝑟Q)
6 prop 6630 . . . . . . 7 (𝑣P → ⟨(1st𝑣), (2nd𝑣)⟩ ∈ P)
7 elprnql 6636 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (1st𝑣)) → 𝑠Q)
86, 7sylan 271 . . . . . 6 ((𝑣P𝑠 ∈ (1st𝑣)) → 𝑠Q)
98adantll 453 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (1st𝑣)) → 𝑠Q)
105, 9genpdflem 6662 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)})
11 elprnqu 6637 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
122, 11sylan 271 . . . . . 6 ((𝑤P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
1312adantlr 454 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (2nd𝑤)) → 𝑟Q)
14 elprnqu 6637 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
156, 14sylan 271 . . . . . 6 ((𝑣P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1615adantll 453 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1713, 16genpdflem 6662 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)})
1810, 17opeq12d 3584 . . 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)}⟩)
1918mpt2eq3ia 5597 . 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 2076 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 101  w3a 896   = wceq 1259  wcel 1409  wrex 2324  {crab 2327  cop 3405  cfv 4929  (class class class)co 5539  cmpt2 5541  1st c1st 5792  2nd c2nd 5793  Qcnq 6435  Pcnp 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-qs 6142  df-ni 6459  df-nqqs 6503  df-inp 6621
This theorem is referenced by:  genipv  6664
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