genpcdl - Intuitionistic Logic Explorer

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

Theorem genpcdl 7833

Description: Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-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)
genpcdl.2	⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑔𝐺ℎ) → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))

**Assertion**
Ref	Expression
genpcdl	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(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 genpcdl**
Step	Hyp	Ref	Expression
1		ltrelnq 7679	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
2	1	brel 4801	. . . . . 6 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
3	2	simpld 112	. . . . 5 ⊢ (𝑥 <_Q 𝑓 → 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	genpelvl 7826	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
7	6	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
8		breq2 4112	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑥 <_Q 𝑓 ↔ 𝑥 <_Q (𝑔𝐺ℎ)))
9	8	biimpd 144	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑥 <_Q 𝑓 → 𝑥 <_Q (𝑔𝐺ℎ)))
10		genpcdl.2	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑔𝐺ℎ) → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))
11	9, 10	sylan9r 410	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) ∧ 𝑓 = (𝑔𝐺ℎ)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))
12	11	exp31 364	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))))
13	12	an4s 592	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (1^st ‘𝐴) ∧ ℎ ∈ (1^st ‘𝐵))) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))))
14	13	impancom 260	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → ((𝑔 ∈ (1^st ‘𝐴) ∧ ℎ ∈ (1^st ‘𝐵)) → (𝑓 = (𝑔𝐺ℎ) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))))
15	14	rexlimdvv 2667	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)))))
16	7, 15	sylbid 150	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)))))
17	16	ex 115	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ Q → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))))
18	3, 17	syl5 32	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 <_Q 𝑓 → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))))
19	18	com34 83	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 <_Q 𝑓 → (𝑥 <_Q 𝑓 → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵))))))
20	19	pm2.43d 50	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 <_Q 𝑓 → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)))))
21	20	com23 78	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 ∃wrex 2521 {crab 2524 ⟨cop 3691 class class class wbr 4108 ‘cfv 5351 (class class class)co 6049 ∈ cmpo 6051 1^st c1st 6331 2^nd c2nd 6332 Qcnq 7594 <_Q cltq 7599 Pcnp 7605

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-id 4413 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-qs 6772 df-ni 7618 df-nqqs 7662 df-ltnqqs 7667 df-inp 7780

This theorem is referenced by: genprndl 7835

W3C validator