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Theorem genpdf 6606
 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 6573 . . . . . . 7 (𝑤P → ⟨(1st𝑤), (2nd𝑤)⟩ ∈ P)
3 elprnql 6579 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (1st𝑤)) → 𝑟Q)
42, 3sylan 267 . . . . . 6 ((𝑤P𝑟 ∈ (1st𝑤)) → 𝑟Q)
54adantlr 446 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (1st𝑤)) → 𝑟Q)
6 prop 6573 . . . . . . 7 (𝑣P → ⟨(1st𝑣), (2nd𝑣)⟩ ∈ P)
7 elprnql 6579 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (1st𝑣)) → 𝑠Q)
86, 7sylan 267 . . . . . 6 ((𝑣P𝑠 ∈ (1st𝑣)) → 𝑠Q)
98adantll 445 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (1st𝑣)) → 𝑠Q)
105, 9genpdflem 6605 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)})
11 elprnqu 6580 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
122, 11sylan 267 . . . . . 6 ((𝑤P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
1312adantlr 446 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (2nd𝑤)) → 𝑟Q)
14 elprnqu 6580 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
156, 14sylan 267 . . . . . 6 ((𝑣P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1615adantll 445 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1713, 16genpdflem 6605 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)})
1810, 17opeq12d 3557 . . 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 5570 . 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 2060 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 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  {crab 2310  ⟨cop 3378  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  1st c1st 5765  2nd c2nd 5766  Qcnq 6378  Pcnp 6389 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-qs 6112  df-ni 6402  df-nqqs 6446  df-inp 6564 This theorem is referenced by:  genipv  6607
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