recexprlempr - Intuitionistic Logic Explorer

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

Theorem recexprlempr 7912

Description: 𝐵 is a positive real. Lemma for recexpr 7918. (Contributed by Jim Kingdon, 27-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlempr	⊢ (𝐴 ∈ P → 𝐵 ∈ P)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦

Proof of Theorem recexprlempr Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemm 7904	. . 3 ⊢ (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵)))
3		ltrelnq 7645	. . . . . . . . . . 11 ⊢ <_Q ⊆ (Q × Q)
4	3	brel 4784	. . . . . . . . . 10 ⊢ (𝑥 <_Q 𝑦 → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
5	4	simpld 112	. . . . . . . . 9 ⊢ (𝑥 <_Q 𝑦 → 𝑥 ∈ Q)
6	5	adantr 276	. . . . . . . 8 ⊢ ((𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
7	6	exlimiv 1647	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
8	7	abssi 3303	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
9		nqex 7643	. . . . . . 7 ⊢ Q ∈ V
10	9	elpw2 4252	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ 𝒫 Q ↔ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q)
11	8, 10	mpbir 146	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ 𝒫 Q
12	3	brel 4784	. . . . . . . . . 10 ⊢ (𝑦 <_Q 𝑥 → (𝑦 ∈ Q ∧ 𝑥 ∈ Q))
13	12	simprd 114	. . . . . . . . 9 ⊢ (𝑦 <_Q 𝑥 → 𝑥 ∈ Q)
14	13	adantr 276	. . . . . . . 8 ⊢ ((𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
15	14	exlimiv 1647	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
16	15	abssi 3303	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
17	9	elpw2 4252	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ 𝒫 Q ↔ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q)
18	16, 17	mpbir 146	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ 𝒫 Q
19		opelxpi 4763	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ 𝒫 Q ∧ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ 𝒫 Q) → ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩ ∈ (𝒫 Q × 𝒫 Q))
20	11, 18, 19	mp2an 426	. . . 4 ⊢ ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩ ∈ (𝒫 Q × 𝒫 Q)
21	1, 20	eqeltri 2304	. . 3 ⊢ 𝐵 ∈ (𝒫 Q × 𝒫 Q)
22	2, 21	jctil 312	. 2 ⊢ (𝐴 ∈ P → (𝐵 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵))))
23	1	recexprlemrnd 7909	. . 3 ⊢ (𝐴 ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))))
24	1	recexprlemdisj 7910	. . 3 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐵) ∧ 𝑞 ∈ (2^nd ‘𝐵)))
25	1	recexprlemloc 7911	. . 3 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐵) ∨ 𝑟 ∈ (2^nd ‘𝐵))))
26	23, 24, 25	3jca 1204	. 2 ⊢ (𝐴 ∈ P → ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐵) ∧ 𝑞 ∈ (2^nd ‘𝐵)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐵) ∨ 𝑟 ∈ (2^nd ‘𝐵)))))
27		elnp1st2nd 7756	. 2 ⊢ (𝐵 ∈ P ↔ ((𝐵 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐵) ∧ 𝑞 ∈ (2^nd ‘𝐵)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐵) ∨ 𝑟 ∈ (2^nd ‘𝐵))))))
28	22, 26, 27	sylanbrc 417	1 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 ∧ w3a 1005 = wceq 1398 ∃wex 1541 ∈ wcel 2202 {cab 2217 ∀wral 2511 ∃wrex 2512 ⊆ wss 3201 𝒫 cpw 3656 ⟨cop 3676 class class class wbr 4093 × cxp 4729 ‘cfv 5333 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7560 *_Qcrq 7564 <_Q cltq 7565 Pcnp 7571

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-1o 6625 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7584 df-pli 7585 df-mi 7586 df-lti 7587 df-plpq 7624 df-mpq 7625 df-enq 7627 df-nqqs 7628 df-plqqs 7629 df-mqqs 7630 df-1nqqs 7631 df-rq 7632 df-ltnqqs 7633 df-inp 7746

This theorem is referenced by: recexprlem1ssl 7913 recexprlem1ssu 7914 recexprlemss1l 7915 recexprlemss1u 7916 recexprlemex 7917 recexpr 7918

W3C validator