genpcuu - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > genpcuu		GIF version

Theorem genpcuu 7352

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 7197	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
2	1	brel 4599	. . . . . 6 ⊢ (𝑓 <_Q 𝑥 → (𝑓 ∈ Q ∧ 𝑥 ∈ Q))
3	2	simprd 113	. . . . 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 7345	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
7	6	adantr 274	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
8		breq1 3940	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <_Q 𝑥 ↔ (𝑔𝐺ℎ) <_Q 𝑥))
9	8	biimpd 143	. . . . . . . . . . . 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 408	. . . . . . . . . . 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 362	. . . . . . . . . 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 578	. . . . . . . . 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 258	. . . . . . . 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 2559	. . . . . . 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 149	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (2^nd ‘(𝐴𝐹𝐵)) → (𝑓 <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴𝐹𝐵)))))
17	16	ex 114	. . . . 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 103 ↔ wb 104 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 {crab 2421 ⟨cop 3535 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ∈ cmpo 5784 1^st c1st 6044 2^nd c2nd 6045 Qcnq 7112 <_Q cltq 7117 Pcnp 7123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-qs 6443 df-ni 7136 df-nqqs 7180 df-ltnqqs 7185 df-inp 7298

This theorem is referenced by: genprndu 7354

W3C validator