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Theorem ltexprlemlol 6700

Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)

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

**Assertion**
Ref	Expression
ltexprlemlol	⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) → 𝑞 ∈ (1^st ‘𝐶)))

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

**Proof of Theorem ltexprlemlol**
Step	Hyp	Ref	Expression
1		simplr 482	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → 𝑞 ∈ Q)
2		simprrr 492	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))
3	2	simpld 105	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → 𝑦 ∈ (2^nd ‘𝐴))
4		simprl 483	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → 𝑞 <_Q 𝑟)
5		simpll 481	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → 𝐴<_P 𝐵)
6		ltrelpr 6603	. . . . . . . . . . . 12 ⊢ <_P ⊆ (P × P)
7	6	brel 4392	. . . . . . . . . . 11 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
8	7	simpld 105	. . . . . . . . . 10 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
9		prop 6573	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
10		elprnqu 6580	. . . . . . . . . . 11 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
11	9, 10	sylan 267	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
12	8, 11	sylan 267	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
13	5, 3, 12	syl2anc 391	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → 𝑦 ∈ Q)
14		ltanqi 6500	. . . . . . . 8 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑦 ∈ Q) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
15	4, 13, 14	syl2anc 391	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
16	7	simprd 107	. . . . . . . . 9 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
17	5, 16	syl 14	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → 𝐵 ∈ P)
18	2	simprd 107	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))
19		prop 6573	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
20		prcdnql 6582	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)) → ((𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟) → (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))
21	19, 20	sylan 267	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)) → ((𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟) → (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))
22	17, 18, 21	syl2anc 391	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → ((𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟) → (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)))
23	15, 22	mpd 13	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))
24	1, 3, 23	jca32 293	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → (𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
25	24	eximi 1491	. . . 4 ⊢ (∃𝑦((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) → ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
26		ltexprlem.1	. . . . . . . . . 10 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩
27	26	ltexprlemell 6696	. . . . . . . . 9 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
28		19.42v 1786	. . . . . . . . 9 ⊢ (∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
29	27, 28	bitr4i 176	. . . . . . . 8 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
30	29	anbi2i 430	. . . . . . 7 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) ↔ (𝑞 <_Q 𝑟 ∧ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
31		19.42v 1786	. . . . . . 7 ⊢ (∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))) ↔ (𝑞 <_Q 𝑟 ∧ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
32	30, 31	bitr4i 176	. . . . . 6 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) ↔ ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵)))))
33	32	anbi2i 430	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) ↔ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))))
34		19.42v 1786	. . . . 5 ⊢ (∃𝑦((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))) ↔ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))))
35	33, 34	bitr4i 176	. . . 4 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) ↔ ∃𝑦((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))))
36	26	ltexprlemell 6696	. . . . 5 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
37		19.42v 1786	. . . . 5 ⊢ (∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
38	36, 37	bitr4i 176	. . . 4 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
39	25, 35, 38	3imtr4i 190	. . 3 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) → 𝑞 ∈ (1^st ‘𝐶))
40	39	ex 108	. 2 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) → 𝑞 ∈ (1^st ‘𝐶)))
41	40	rexlimdvw 2436	1 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) → 𝑞 ∈ (1^st ‘𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 {crab 2310 ⟨cop 3378 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 1^st c1st 5765 2^nd c2nd 5766 Qcnq 6378 +_Q cplq 6380 <_Q cltq 6383 Pcnp 6389 <_P cltp 6393

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-ltnqqs 6451 df-inp 6564 df-iltp 6568

This theorem is referenced by: ltexprlemrnd 6703

W3C validator