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Theorem genpelvu 7435
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 7431 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩)
43fveq2d 5474 . . . . 5 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩))
5 nqex 7285 . . . . . . 7 Q ∈ V
65rabex 4110 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)} ∈ V
75rabex 4110 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ∈ V
86, 7op2nd 6097 . . . . 5 (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}
94, 8eqtrdi 2206 . . . 4 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)})
109eleq2d 2227 . . 3 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}))
11 elrabi 2865 . . 3 (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} → 𝐶Q)
1210, 11syl6bi 162 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝐶Q))
13 prop 7397 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
14 elprnqu 7404 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
1513, 14sylan 281 . . . . . 6 ((𝐴P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
16 prop 7397 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
17 elprnqu 7404 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∈ (2nd𝐵)) → Q)
1816, 17sylan 281 . . . . . 6 ((𝐵P ∈ (2nd𝐵)) → Q)
192caovcl 5977 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
2015, 18, 19syl2an 287 . . . . 5 (((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
2120an4s 578 . . . 4 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
22 eleq1 2220 . . . 4 (𝐶 = (𝑔𝐺) → (𝐶Q ↔ (𝑔𝐺) ∈ Q))
2321, 22syl5ibrcom 156 . . 3 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝐶 = (𝑔𝐺) → 𝐶Q))
2423rexlimdvva 2582 . 2 ((𝐴P𝐵P) → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺) → 𝐶Q))
25 eqeq1 2164 . . . . . 6 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
26252rexbidv 2482 . . . . 5 (𝑓 = 𝐶 → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2726elrab3 2869 . . . 4 (𝐶Q → (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2810, 27sylan9bb 458 . . 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 695 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 963   = wceq 1335  wcel 2128  wrex 2436  {crab 2439  cop 3564  cfv 5172  (class class class)co 5826  cmpo 5828  1st c1st 6088  2nd c2nd 6089  Qcnq 7202  Pcnp 7213
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-qs 6488  df-ni 7226  df-nqqs 7270  df-inp 7388
This theorem is referenced by:  genppreclu  7437  genpcuu  7442  genprndu  7444  genpdisj  7445  genpassu  7447  addnqprlemru  7480  mulnqprlemru  7496  distrlem1pru  7505  distrlem5pru  7509  1idpru  7513  ltexprlemfu  7533  recexprlem1ssu  7556  recexprlemss1u  7558  cauappcvgprlemladdfu  7576  caucvgprlemladdfu  7599
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