genpcuu - Intuitionistic Logic Explorer

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Theorem genpcuu 7640

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 7485	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
2	1	brel 4731	. . . . . 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 7633	. . . . . . . 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 4050	. . . . . . . . . . . . 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 588	. . . . . . . . 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 2631	. . . . . . 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 981 = wceq 1373 ∈ wcel 2177 ∃wrex 2486 {crab 2489 ⟨cop 3637 class class class wbr 4047 ‘cfv 5276 (class class class)co 5951 ∈ cmpo 5953 1^st c1st 6231 2^nd c2nd 6232 Qcnq 7400 <_Q cltq 7405 Pcnp 7411

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-id 4344 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-qs 6633 df-ni 7424 df-nqqs 7468 df-ltnqqs 7473 df-inp 7586

This theorem is referenced by: genprndu 7642

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