genpcdl - Intuitionistic Logic Explorer

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Theorem genpcdl 7460

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 7306	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
2	1	brel 4656	. . . . . 6 ⊢ (𝑥 <_Q 𝑓 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q))
3	2	simpld 111	. . . . 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 7453	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
7	6	adantr 274	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
8		breq2 3986	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑥 <_Q 𝑓 ↔ 𝑥 <_Q (𝑔𝐺ℎ)))
9	8	biimpd 143	. . . . . . . . . . . 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 408	. . . . . . . . . . 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 362	. . . . . . . . . 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 578	. . . . . . . . 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 258	. . . . . . . 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 2590	. . . . . . 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 149	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (1^st ‘(𝐴𝐹𝐵)) → (𝑥 <_Q 𝑓 → 𝑥 ∈ (1^st ‘(𝐴𝐹𝐵)))))
17	16	ex 114	. . . . 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 103 ↔ wb 104 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ∃wrex 2445 {crab 2448 ⟨cop 3579 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 ∈ cmpo 5844 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 <_Q cltq 7226 Pcnp 7232

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-qs 6507 df-ni 7245 df-nqqs 7289 df-ltnqqs 7294 df-inp 7407

This theorem is referenced by: genprndl 7462

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