ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genpcdl GIF version

Theorem genpcdl 6675
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 6521 . . . . . . 7 <Q ⊆ (Q × Q)
21brel 4420 . . . . . 6 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
32simpld 109 . . . . 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 6668 . . . . . . . 8 ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)))
76adantr 265 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)))
8 breq2 3796 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 <Q (𝑔𝐺)))
98biimpd 136 . . . . . . . . . . . 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 396 . . . . . . . . . . 11 (((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) ∧ 𝑓 = (𝑔𝐺)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))
1211exp31 350 . . . . . . . . . 10 (((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1312an4s 530 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1413impancom 251 . . . . . . . 8 (((𝐴P𝐵P) ∧ 𝑥Q) → ((𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵)) → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1514rexlimdvv 2456 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
167, 15sylbid 143 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
1716ex 112 . . . . 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 81 . . 3 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
2019pm2.43d 48 . 2 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
2120com23 76 1 ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(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 3406   class class class wbr 3792  cfv 4930  (class class class)co 5540  cmpt2 5542  1st c1st 5793  2nd c2nd 5794  Qcnq 6436   <Q cltq 6441  Pcnp 6447
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 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
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 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-qs 6143  df-ni 6460  df-nqqs 6504  df-ltnqqs 6509  df-inp 6622
This theorem is referenced by:  genprndl  6677
  Copyright terms: Public domain W3C validator