genpcuu - Intuitionistic Logic Explorer

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Theorem genpcuu 7675

Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)

**Hypotheses**
Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genpcuu.2	⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2^nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))

**Assertion**
Ref	Expression
genpcuu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵)))))

Distinct variable groups: 𝑥,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣,𝐴 𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣 𝑥,𝐺,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣 𝑓,𝐹,𝑔,ℎ Allowed substitution hints: 𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

**Proof of Theorem genpcuu**
Step	Hyp	Ref	Expression
1		ltrelnq 7520	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
2	1	brel 4748	. . . . . 6 ⊢ (𝑓 <_Q 𝑥 → (𝑓 ∈ Q ∧ 𝑥 ∈ Q))
3	2	simprd 114	. . . . 5 ⊢ (𝑓 <_Q 𝑥 → 𝑥 ∈ Q)
4		genpelvl.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
5		genpelvl.2	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
6	4, 5	genpelvu 7668	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
7	6	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
8		breq1 4065	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <_Q 𝑥 ↔ (𝑔𝐺ℎ) <_Q 𝑥))
9	8	biimpd 144	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <_Q 𝑥 → (𝑔𝐺ℎ) <_Q 𝑥))
10		genpcuu.2	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2^nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))
11	9, 10	sylan9r 410	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ P ∧ 𝑔 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2^nd ‘𝐵))) ∧ 𝑥 ∈ Q) ∧ 𝑓 = (𝑔𝐺ℎ)) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))
12	11	exp31 364	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2^nd ‘𝐵))) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))))
13	12	an4s 590	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (2^nd ‘𝐴) ∧ ℎ ∈ (2^nd ‘𝐵))) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))))
14	13	impancom 260	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → ((𝑔 ∈ (2^nd ‘𝐴) ∧ ℎ ∈ (2^nd ‘𝐵)) → (𝑓 = (𝑔𝐺ℎ) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))))
15	14	rexlimdvv 2635	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵)))))
16	7, 15	sylbid 150	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵)))))
17	16	ex 115	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ Q → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))))
18	3, 17	syl5 32	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 <_Q 𝑥 → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))))
19	18	com34 83	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 <_Q 𝑥 → (𝑓 <_Q 𝑥 → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵))))))
20	19	pm2.43d 50	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 <_Q 𝑥 → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵)))))
21	20	com23 78	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵)))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ∃wrex 2489 {crab 2492 ⟨cop 3649 class class class wbr 4062 ‘cfv 5294 (class class class)co 5974 ∈ cmpo 5976 1^st c1st 6254 2^nd c2nd 6255 Qcnq 7435 <_Q cltq 7440 Pcnp 7446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657

This theorem depends on definitions: df-bi 117 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-ral 2493 df-rex 2494 df-reu 2495 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-id 4361 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-qs 6656 df-ni 7459 df-nqqs 7503 df-ltnqqs 7508 df-inp 7621

This theorem is referenced by: genprndu 7677

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