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Theorem genpelvu 7514
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 7510 . . . . . 6 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = ⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩)
43fveq2d 5521 . . . . 5 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩))
5 nqex 7364 . . . . . . 7 Q ∈ V
65rabex 4149 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)} ∈ V
75rabex 4149 . . . . . 6 {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ∈ V
86, 7op2nd 6150 . . . . 5 (2nd ‘⟨{𝑓Q ∣ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)}, {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}⟩) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}
94, 8eqtrdi 2226 . . . 4 ((𝐴P𝐵P) → (2nd ‘(𝐴𝐹𝐵)) = {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)})
109eleq2d 2247 . . 3 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)}))
11 elrabi 2892 . . 3 (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} → 𝐶Q)
1210, 11biimtrdi 163 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝐶Q))
13 prop 7476 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
14 elprnqu 7483 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
1513, 14sylan 283 . . . . . 6 ((𝐴P𝑔 ∈ (2nd𝐴)) → 𝑔Q)
16 prop 7476 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
17 elprnqu 7483 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∈ (2nd𝐵)) → Q)
1816, 17sylan 283 . . . . . 6 ((𝐵P ∈ (2nd𝐵)) → Q)
192caovcl 6031 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
2015, 18, 19syl2an 289 . . . . 5 (((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
2120an4s 588 . . . 4 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝑔𝐺) ∈ Q)
22 eleq1 2240 . . . 4 (𝐶 = (𝑔𝐺) → (𝐶Q ↔ (𝑔𝐺) ∈ Q))
2321, 22syl5ibrcom 157 . . 3 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝐶 = (𝑔𝐺) → 𝐶Q))
2423rexlimdvva 2602 . 2 ((𝐴P𝐵P) → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺) → 𝐶Q))
25 eqeq1 2184 . . . . . 6 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
26252rexbidv 2502 . . . . 5 (𝑓 = 𝐶 → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2726elrab3 2896 . . . 4 (𝐶Q → (𝐶 ∈ {𝑓Q ∣ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)} ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2810, 27sylan9bb 462 . . 3 (((𝐴P𝐵P) ∧ 𝐶Q) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
2928ex 115 . 2 ((𝐴P𝐵P) → (𝐶Q → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺))))
3012, 24, 29pm5.21ndd 705 1 ((𝐴P𝐵P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝐶 = (𝑔𝐺)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wrex 2456  {crab 2459  cop 3597  cfv 5218  (class class class)co 5877  cmpo 5879  1st c1st 6141  2nd c2nd 6142  Qcnq 7281  Pcnp 7292
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-qs 6543  df-ni 7305  df-nqqs 7349  df-inp 7467
This theorem is referenced by:  genppreclu  7516  genpcuu  7521  genprndu  7523  genpdisj  7524  genpassu  7526  addnqprlemru  7559  mulnqprlemru  7575  distrlem1pru  7584  distrlem5pru  7588  1idpru  7592  ltexprlemfu  7612  recexprlem1ssu  7635  recexprlemss1u  7637  cauappcvgprlemladdfu  7655  caucvgprlemladdfu  7678
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