recexprlemopl - Intuitionistic Logic Explorer

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

Theorem recexprlemopl 7758

Description: The lower cut of 𝐵 is open. Lemma for recexpr 7771. (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 7755	. . 3 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
3		ltbtwnnqq 7548	. . . . . 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 1614	. . . . . . . . . 10 ⊢ ((𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)))
7	1	recexprlemell 7755	. . . . . . . . . 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 2608	. . . . 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 1622	. . 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 1023	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 981 = wceq 1373 ∃wex 1516 ∈ wcel 2177 {cab 2192 ∃wrex 2486 ⟨cop 3641 class class class wbr 4051 ‘cfv 5280 1^st c1st 6237 2^nd c2nd 6238 Qcnq 7413 *_Qcrq 7417 <_Q cltq 7418 Pcnp 7424

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-nul 4178 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-iinf 4644

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-tr 4151 df-eprel 4344 df-id 4348 df-po 4351 df-iso 4352 df-iord 4421 df-on 4423 df-suc 4426 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-ov 5960 df-oprab 5961 df-mpo 5962 df-1st 6239 df-2nd 6240 df-recs 6404 df-irdg 6469 df-1o 6515 df-oadd 6519 df-omul 6520 df-er 6633 df-ec 6635 df-qs 6639 df-ni 7437 df-pli 7438 df-mi 7439 df-lti 7440 df-plpq 7477 df-mpq 7478 df-enq 7480 df-nqqs 7481 df-plqqs 7482 df-mqqs 7483 df-1nqqs 7484 df-rq 7485 df-ltnqqs 7486

This theorem is referenced by: recexprlemrnd 7762

W3C validator