genpcdl - Intuitionistic Logic Explorer

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Theorem genpcdl 7520

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 7366	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
2	1	brel 4680	. . . . . 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 7513	. . . . . . . 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 4009	. . . . . . . . . . . . 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 588	. . . . . . . . 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 2601	. . . . . . 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 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 {crab 2459 ⟨cop 3597 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 ∈ cmpo 5879 1^st c1st 6141 2^nd c2nd 6142 Qcnq 7281 <_Q cltq 7286 Pcnp 7292

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-qs 6543 df-ni 7305 df-nqqs 7349 df-ltnqqs 7354 df-inp 7467

This theorem is referenced by: genprndl 7522

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