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Theorem genpdf 7538
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 7505 . . . . . . 7 (𝑤P → ⟨(1st𝑤), (2nd𝑤)⟩ ∈ P)
3 elprnql 7511 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (1st𝑤)) → 𝑟Q)
42, 3sylan 283 . . . . . 6 ((𝑤P𝑟 ∈ (1st𝑤)) → 𝑟Q)
54adantlr 477 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (1st𝑤)) → 𝑟Q)
6 prop 7505 . . . . . . 7 (𝑣P → ⟨(1st𝑣), (2nd𝑣)⟩ ∈ P)
7 elprnql 7511 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (1st𝑣)) → 𝑠Q)
86, 7sylan 283 . . . . . 6 ((𝑣P𝑠 ∈ (1st𝑣)) → 𝑠Q)
98adantll 476 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (1st𝑣)) → 𝑠Q)
105, 9genpdflem 7537 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑤) ∧ 𝑠 ∈ (1st𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (1st𝑤)∃𝑠 ∈ (1st𝑣)𝑞 = (𝑟𝐺𝑠)})
11 elprnqu 7512 . . . . . . 7 ((⟨(1st𝑤), (2nd𝑤)⟩ ∈ P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
122, 11sylan 283 . . . . . 6 ((𝑤P𝑟 ∈ (2nd𝑤)) → 𝑟Q)
1312adantlr 477 . . . . 5 (((𝑤P𝑣P) ∧ 𝑟 ∈ (2nd𝑤)) → 𝑟Q)
14 elprnqu 7512 . . . . . . 7 ((⟨(1st𝑣), (2nd𝑣)⟩ ∈ P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
156, 14sylan 283 . . . . . 6 ((𝑣P𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1615adantll 476 . . . . 5 (((𝑤P𝑣P) ∧ 𝑠 ∈ (2nd𝑣)) → 𝑠Q)
1713, 16genpdflem 7537 . . . 4 ((𝑤P𝑣P) → {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑤) ∧ 𝑠 ∈ (2nd𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞Q ∣ ∃𝑟 ∈ (2nd𝑤)∃𝑠 ∈ (2nd𝑣)𝑞 = (𝑟𝐺𝑠)})
1810, 17opeq12d 3801 . . 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 5962 . 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 2210 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 104  w3a 980   = wceq 1364  wcel 2160  wrex 2469  {crab 2472  cop 3610  cfv 5235  (class class class)co 5897  cmpo 5899  1st c1st 6164  2nd c2nd 6165  Qcnq 7310  Pcnp 7321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-qs 6566  df-ni 7334  df-nqqs 7378  df-inp 7496
This theorem is referenced by:  genipv  7539
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