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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
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