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Theorem genpcuu 6824
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 6669 . . . . . . 7 <Q ⊆ (Q × Q)
21brel 4438 . . . . . 6 (𝑓 <Q 𝑥 → (𝑓Q𝑥Q))
32simprd 112 . . . . 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 6817 . . . . . . . 8 ((𝐴P𝐵P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)))
76adantr 270 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺)))
8 breq1 3808 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺) <Q 𝑥))
98biimpd 142 . . . . . . . . . . . 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 402 . . . . . . . . . . 11 (((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) ∧ 𝑓 = (𝑔𝐺)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
1211exp31 356 . . . . . . . . . 10 (((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
1312an4s 553 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
1413impancom 256 . . . . . . . 8 (((𝐴P𝐵P) ∧ 𝑥Q) → ((𝑔 ∈ (2nd𝐴) ∧ ∈ (2nd𝐵)) → (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
1514rexlimdvv 2488 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (∃𝑔 ∈ (2nd𝐴)∃ ∈ (2nd𝐵)𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
167, 15sylbid 148 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
1716ex 113 . . . . 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 82 . . 3 ((𝐴P𝐵P) → (𝑓 <Q 𝑥 → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
2019pm2.43d 49 . 2 ((𝐴P𝐵P) → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
2120com23 77 1 ((𝐴P𝐵P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wrex 2354  {crab 2357  cop 3419   class class class wbr 3805  cfv 4952  (class class class)co 5563  cmpt2 5565  1st c1st 5816  2nd c2nd 5817  Qcnq 6584   <Q cltq 6589  Pcnp 6595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-qs 6199  df-ni 6608  df-nqqs 6652  df-ltnqqs 6657  df-inp 6770
This theorem is referenced by:  genprndu  6826
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