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Theorem recexprlemopl 7823

Description: The lower cut of 𝐵 is open. Lemma for recexpr 7836. (Contributed by Jim Kingdon, 28-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
recexprlemopl	⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))

Distinct variable groups: 𝑟,𝑞,𝑥,𝑦,𝐴 𝐵,𝑞,𝑟,𝑥,𝑦

**Proof of Theorem recexprlemopl**
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	recexprlemell 7820	. . 3 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
3		ltbtwnnqq 7613	. . . . . 6 ⊢ (𝑞 <_Q 𝑦 ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦))
4	3	biimpi 120	. . . . 5 ⊢ (𝑞 <_Q 𝑦 → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦))
5		simpll 527	. . . . . . . 8 ⊢ (((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑞 <_Q 𝑟)
6		19.8a 1636	. . . . . . . . . 10 ⊢ ((𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)))
7	1	recexprlemell 7820	. . . . . . . . . 10 ⊢ (𝑟 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑟 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
8	6, 7	sylibr 134	. . . . . . . . 9 ⊢ ((𝑟 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑟 ∈ (1^st ‘𝐵))
9	8	adantll 476	. . . . . . . 8 ⊢ (((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑟 ∈ (1^st ‘𝐵))
10	5, 9	jca 306	. . . . . . 7 ⊢ (((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
11	10	expcom 116	. . . . . 6 ⊢ ((*_Q‘𝑦) ∈ (2^nd ‘𝐴) → ((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) → (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))))
12	11	reximdv 2631	. . . . 5 ⊢ ((*_Q‘𝑦) ∈ (2^nd ‘𝐴) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))))
13	4, 12	mpan9 281	. . . 4 ⊢ ((𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
14	13	exlimiv 1644	. . 3 ⊢ (∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
15	2, 14	sylbi 121	. 2 ⊢ (𝑞 ∈ (1^st ‘𝐵) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))
16	15	3ad2ant3 1044	1 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1002 = wceq 1395 ∃wex 1538 ∈ wcel 2200 {cab 2215 ∃wrex 2509 ⟨cop 3669 class class class wbr 4083 ‘cfv 5318 1^st c1st 6290 2^nd c2nd 6291 Qcnq 7478 *_Qcrq 7482 <_Q cltq 7483 Pcnp 7489

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-eprel 4380 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-1o 6568 df-oadd 6572 df-omul 6573 df-er 6688 df-ec 6690 df-qs 6694 df-ni 7502 df-pli 7503 df-mi 7504 df-lti 7505 df-plpq 7542 df-mpq 7543 df-enq 7545 df-nqqs 7546 df-plqqs 7547 df-mqqs 7548 df-1nqqs 7549 df-rq 7550 df-ltnqqs 7551

This theorem is referenced by: recexprlemrnd 7827

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