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Theorem genpdf 7121
 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 7088 . . . . . . 7 (𝑤P → ⟨(1st𝑤), (2nd𝑤)⟩ ∈ P)
3 elprnql 7094 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (1st𝑤)) → 𝑟Q)
42, 3sylan 278 . . . . . 6 ((𝑤P𝑟 ∈ (1st𝑤)) → 𝑟Q)
54adantlr 462 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (1st𝑤)) → 𝑟Q)
6 prop 7088 . . . . . . 7 (𝑣P → ⟨(1st𝑣), (2nd𝑣)⟩ ∈ P)
7 elprnql 7094 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (1st𝑣)) → 𝑠Q)
86, 7sylan 278 . . . . . 6 ((𝑣P𝑠 ∈ (1st𝑣)) → 𝑠Q)
98adantll 461 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (1st𝑣)) → 𝑠Q)
105, 9genpdflem 7120 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)})
11 elprnqu 7095 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
122, 11sylan 278 . . . . . 6 ((𝑤P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
1312adantlr 462 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (2nd𝑤)) → 𝑟Q)
14 elprnqu 7095 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
156, 14sylan 278 . . . . . 6 ((𝑣P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1615adantll 461 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1713, 16genpdflem 7120 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)})
1810, 17opeq12d 3636 . . 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 5728 . 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 2109 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 925   = wceq 1290   ∈ wcel 1439  ∃wrex 2361  {crab 2364  ⟨cop 3453  ‘cfv 5028  (class class class)co 5666   ↦ cmpt2 5668  1st c1st 5923  2nd c2nd 5924  Qcnq 6893  Pcnp 6904 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-iinf 4416 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-qs 6312  df-ni 6917  df-nqqs 6961  df-inp 7079 This theorem is referenced by:  genipv  7122
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