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Theorem genpcdl 6574
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 6420 . . . . . . 7 <Q ⊆ (Q × Q)
21brel 4355 . . . . . 6 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
32simpld 105 . . . . 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 6567 . . . . . . . 8 ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)))
76adantr 261 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺)))
8 breq2 3765 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 <Q (𝑔𝐺)))
98biimpd 132 . . . . . . . . . . . 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 390 . . . . . . . . . . 11 (((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) ∧ 𝑓 = (𝑔𝐺)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))
1211exp31 346 . . . . . . . . . 10 (((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1312an4s 522 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵))) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1413impancom 247 . . . . . . . 8 (((𝐴P𝐵P) ∧ 𝑥Q) → ((𝑔 ∈ (1st𝐴) ∧ ∈ (1st𝐵)) → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1514rexlimdvv 2436 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (∃𝑔 ∈ (1st𝐴)∃ ∈ (1st𝐵)𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
167, 15sylbid 139 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
1716ex 108 . . . . 5 ((𝐴P𝐵P) → (𝑥Q → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
183, 17syl5 28 . . . 4 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
1918com34 77 . . 3 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))))
2019pm2.43d 44 . 2 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
2120com23 72 1 ((𝐴P𝐵P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  wrex 2304  {crab 2307  cop 3375   class class class wbr 3761  cfv 4865  (class class class)co 5475  cmpt2 5477  1st c1st 5728  2nd c2nd 5729  Qcnq 6335   <Q cltq 6340  Pcnp 6346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-qs 6075  df-ni 6359  df-nqqs 6403  df-ltnqqs 6408  df-inp 6521
This theorem is referenced by:  genprndl  6576
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