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Theorem genpcdl 6979
Description: Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-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)
genpcdl.2 ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))
Assertion
Ref Expression
genpcdl ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(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 genpcdl
StepHypRef Expression
1 ltrelnq 6825 . . . . . . 7 <Q ⊆ (Q × Q)
21brel 4446 . . . . . 6 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
32simpld 110 . . . . 5 (𝑥 <Q 𝑓𝑥Q)
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, 5genpelvl 6972 . . . . . . . 8 ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)))
76adantr 270 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)))
8 breq2 3815 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 <Q (𝑔𝐺)))
98biimpd 142 . . . . . . . . . . . 12 (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 <Q (𝑔𝐺)))
10 genpcdl.2 . . . . . . . . . . . 12 ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))
119, 10sylan9r 402 . . . . . . . . . . 11 (((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) ∧ 𝑓 = (𝑔𝐺)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))
1211exp31 356 . . . . . . . . . 10 (((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1312an4s 553 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1413impancom 256 . . . . . . . 8 (((𝐴P𝐵P) ∧ 𝑥Q) → ((𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵)) → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1514rexlimdvv 2489 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
167, 15sylbid 148 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
1716ex 113 . . . . 5 ((𝐴P𝐵P) → (𝑥Q → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
183, 17syl5 32 . . . 4 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1918com34 82 . . 3 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
2019pm2.43d 49 . 2 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
2120com23 77 1 ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(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 3425   class class class wbr 3811  cfv 4967  (class class class)co 5589  cmpt2 5591  1st c1st 5842  2nd c2nd 5843  Qcnq 6740   <Q cltq 6745  Pcnp 6751
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 3919  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
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 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-qs 6226  df-ni 6764  df-nqqs 6808  df-ltnqqs 6813  df-inp 6926
This theorem is referenced by:  genprndl  6981
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