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

Theorem genpelvu 7475
Description: Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpelvu ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑔,,𝑤,𝑣   𝑔,𝐹   𝐶,𝑔,
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,)

Proof of Theorem genpelvu
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . 7 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2 genpelvl.2 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genipv 7471 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩)
43fveq2d 5500 . . . . 5 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩))
5 nqex 7325 . . . . . . 7 Q ∈ V
65rabex 4133 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)} ∈ V
75rabex 4133 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ∈ V
86, 7op2nd 6126 . . . . 5 (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}
94, 8eqtrdi 2219 . . . 4 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)})
109eleq2d 2240 . . 3 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}))
11 elrabi 2883 . . 3 (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} → 𝐶Q)
1210, 11syl6bi 162 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝐶Q))
13 prop 7437 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
14 elprnqu 7444 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
1513, 14sylan 281 . . . . . 6 ((𝐴P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
16 prop 7437 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
17 elprnqu 7444 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∈ (2nd𝐵)) → Q)
1816, 17sylan 281 . . . . . 6 ((𝐵P ∈ (2nd𝐵)) → Q)
192caovcl 6007 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
2015, 18, 19syl2an 287 . . . . 5 (((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
2120an4s 583 . . . 4 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
22 eleq1 2233 . . . 4 (𝐶 = (𝑔𝐺) → (𝐶Q ↔ (𝑔𝐺) ∈ Q))
2321, 22syl5ibrcom 156 . . 3 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝐶 = (𝑔𝐺) → 𝐶Q))
2423rexlimdvva 2595 . 2 ((𝐴P𝐵P) → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺) → 𝐶Q))
25 eqeq1 2177 . . . . . 6 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
26252rexbidv 2495 . . . . 5 (𝑓 = 𝐶 → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2726elrab3 2887 . . . 4 (𝐶Q → (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2810, 27sylan9bb 459 . . 3 (((𝐴P𝐵P) ∧ 𝐶Q) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2928ex 114 . 2 ((𝐴P𝐵P) → (𝐶Q → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺))))
3012, 24, 29pm5.21ndd 700 1 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  wrex 2449  {crab 2452  cop 3586  cfv 5198  (class class class)co 5853  cmpo 5855  1st c1st 6117  2nd c2nd 6118  Qcnq 7242  Pcnp 7253
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-qs 6519  df-ni 7266  df-nqqs 7310  df-inp 7428
This theorem is referenced by:  genppreclu  7477  genpcuu  7482  genprndu  7484  genpdisj  7485  genpassu  7487  addnqprlemru  7520  mulnqprlemru  7536  distrlem1pru  7545  distrlem5pru  7549  1idpru  7553  ltexprlemfu  7573  recexprlem1ssu  7596  recexprlemss1u  7598  cauappcvgprlemladdfu  7616  caucvgprlemladdfu  7639
  Copyright terms: Public domain W3C validator