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Theorem recexprlemopl 7692

Description: The lower cut of 𝐵 is open. Lemma for recexpr 7705. (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 7689	. . 3 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
3		ltbtwnnqq 7482	. . . . . 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 1604	. . . . . . . . . 10 ⊢ ((𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)))
7	1	recexprlemell 7689	. . . . . . . . . 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 2598	. . . . 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 1612	. . 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 1022	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 980 = wceq 1364 ∃wex 1506 ∈ wcel 2167 {cab 2182 ∃wrex 2476 ⟨cop 3625 class class class wbr 4033 ‘cfv 5258 1^st c1st 6196 2^nd c2nd 6197 Qcnq 7347 *_Qcrq 7351 <_Q cltq 7352 Pcnp 7358

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-eprel 4324 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-1o 6474 df-oadd 6478 df-omul 6479 df-er 6592 df-ec 6594 df-qs 6598 df-ni 7371 df-pli 7372 df-mi 7373 df-lti 7374 df-plpq 7411 df-mpq 7412 df-enq 7414 df-nqqs 7415 df-plqqs 7416 df-mqqs 7417 df-1nqqs 7418 df-rq 7419 df-ltnqqs 7420

This theorem is referenced by: recexprlemrnd 7696

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