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Theorem genpcuu 6675
Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-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)
genpcuu.2 ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
Assertion
Ref Expression
genpcuu ((𝐴P𝐵P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣   𝑓,𝐹,𝑔,
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpcuu
StepHypRef Expression
1 ltrelnq 6520 . . . . . . 7 <Q ⊆ (Q × Q)
21brel 4419 . . . . . 6 (𝑓 <Q 𝑥 → (𝑓Q𝑥Q))
32simprd 111 . . . . 5 (𝑓 <Q 𝑥𝑥Q)
4 genpelvl.1 . . . . . . . . 9 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
5 genpelvl.2 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
64, 5genpelvu 6668 . . . . . . . 8 ((𝐴P𝐵P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)))
76adantr 265 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)))
8 breq1 3794 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺) <Q 𝑥))
98biimpd 136 . . . . . . . . . . . 12 (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥 → (𝑔𝐺) <Q 𝑥))
10 genpcuu.2 . . . . . . . . . . . 12 ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔𝐺) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
119, 10sylan9r 396 . . . . . . . . . . 11 (((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) ∧ 𝑓 = (𝑔𝐺)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
1211exp31 350 . . . . . . . . . 10 (((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
1312an4s 530 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
1413impancom 251 . . . . . . . 8 (((𝐴P𝐵P) ∧ 𝑥Q) → ((𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵)) → (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
1514rexlimdvv 2456 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
167, 15sylbid 143 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
1716ex 112 . . . . 5 ((𝐴P𝐵P) → (𝑥Q → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
183, 17syl5 32 . . . 4 ((𝐴P𝐵P) → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
1918com34 81 . . 3 ((𝐴P𝐵P) → (𝑓 <Q 𝑥 → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
2019pm2.43d 48 . 2 ((𝐴P𝐵P) → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
2120com23 76 1 ((𝐴P𝐵P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wrex 2324  {crab 2327  cop 3405   class class class wbr 3791  cfv 4929  (class class class)co 5539  cmpt2 5541  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   <Q cltq 6440  Pcnp 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-qs 6142  df-ni 6459  df-nqqs 6503  df-ltnqqs 6508  df-inp 6621
This theorem is referenced by:  genprndu  6677
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