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Theorem ltexprlemlol 6792

Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 6803. (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 496	. . . . . 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 506	. . . . . . 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 110	. . . . . 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 497	. . . . . . . 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 495	. . . . . . . . 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 6695	. . . . . . . . . . . 12 ⊢ <_P ⊆ (P × P)
7	6	brel 4410	. . . . . . . . . . 11 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
8	7	simpld 110	. . . . . . . . . 10 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
9		prop 6665	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
10		elprnqu 6672	. . . . . . . . . . 11 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
11	9, 10	sylan 277	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
12	8, 11	sylan 277	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
13	5, 3, 12	syl2anc 403	. . . . . . . 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 6592	. . . . . . . 8 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑦 ∈ Q) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
15	4, 13, 14	syl2anc 403	. . . . . . 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 112	. . . . . . . . 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 112	. . . . . . . 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 6665	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
20		prcdnql 6674	. . . . . . . . 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 277	. . . . . . . 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 403	. . . . . . 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 303	. . . . 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 1531	. . . 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 6788	. . . . . . . . 9 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
28		19.42v 1827	. . . . . . . . 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 185	. . . . . . . 8 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
30	29	anbi2i 444	. . . . . . 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 1827	. . . . . . 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 185	. . . . . 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 444	. . . . 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 1827	. . . . 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 185	. . . 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 6788	. . . . 5 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
37		19.42v 1827	. . . . 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 185	. . . 4 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
39	25, 35, 38	3imtr4i 199	. . 3 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) → 𝑞 ∈ (1^st ‘𝐶))
40	39	ex 113	. 2 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) → 𝑞 ∈ (1^st ‘𝐶)))
41	40	rexlimdvw 2480	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 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∃wrex 2349 {crab 2352 ⟨cop 3401 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 1^st c1st 5785 2^nd c2nd 5786 Qcnq 6470 +_Q cplq 6472 <_Q cltq 6475 Pcnp 6481 <_P cltp 6485

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-ltnqqs 6543 df-inp 6656 df-iltp 6660

This theorem is referenced by: ltexprlemrnd 6795

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