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Theorem genpcd 9866
Description: Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
genpcd.2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (𝐴𝐹𝐵)))
Assertion
Ref Expression
genpcd ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑓,𝑔,   𝑓,𝐹,𝑔,
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpcd
StepHypRef Expression
1 ltrelnq 9786 . . . . . . 7 <Q ⊆ (Q × Q)
21brel 5202 . . . . . 6 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
32simpld 474 . . . . 5 (𝑥 <Q 𝑓𝑥Q)
4 genp.1 . . . . . . . . 9 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
5 genp.2 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
64, 5genpelv 9860 . . . . . . . 8 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
76adantr 480 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
8 breq2 4689 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 <Q (𝑔𝐺)))
98biimpd 219 . . . . . . . . . . . 12 (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 <Q (𝑔𝐺)))
10 genpcd.2 . . . . . . . . . . . 12 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (𝐴𝐹𝐵)))
119, 10sylan9r 691 . . . . . . . . . . 11 (((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) ∧ 𝑓 = (𝑔𝐺)) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵)))
1211exp31 629 . . . . . . . . . 10 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵)))))
1312an4s 886 . . . . . . . . 9 (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → (𝑥Q → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵)))))
1413impancom 455 . . . . . . . 8 (((𝐴P𝐵P) ∧ 𝑥Q) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵)))))
1514rexlimdvv 3066 . . . . . . 7 (((𝐴P𝐵P) ∧ 𝑥Q) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵))))
167, 15sylbid 230 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑥Q) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵))))
1716ex 449 . . . . 5 ((𝐴P𝐵P) → (𝑥Q → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵)))))
183, 17syl5 34 . . . 4 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵)))))
1918com34 91 . . 3 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑥 ∈ (𝐴𝐹𝐵)))))
2019pm2.43d 53 . 2 ((𝐴P𝐵P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑥 ∈ (𝐴𝐹𝐵))))
2120com23 86 1 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942   class class class wbr 4685  (class class class)co 6690  cmpt2 6692  Qcnq 9712   <Q cltq 9718  Pcnp 9719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-ni 9732  df-nq 9772  df-ltnq 9778  df-np 9841
This theorem is referenced by:  genpcl  9868
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