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Theorem ltexprlemlol 7600

Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7611. (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 528	. . . . . 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 540	. . . . . . 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 112	. . . . . 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 529	. . . . . . . 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 527	. . . . . . . . 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 7503	. . . . . . . . . . . 12 ⊢ <_P ⊆ (P × P)
7	6	brel 4678	. . . . . . . . . . 11 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
8	7	simpld 112	. . . . . . . . . 10 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
9		prop 7473	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
10		elprnqu 7480	. . . . . . . . . . 11 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
11	9, 10	sylan 283	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
12	8, 11	sylan 283	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
13	5, 3, 12	syl2anc 411	. . . . . . . 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 7400	. . . . . . . 8 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑦 ∈ Q) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
15	4, 13, 14	syl2anc 411	. . . . . . 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 114	. . . . . . . . 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 114	. . . . . . . 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 7473	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
20		prcdnql 7482	. . . . . . . . 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 283	. . . . . . . 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 411	. . . . . . 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 310	. . . . 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 1600	. . . 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 7596	. . . . . . . . 9 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
28		19.42v 1906	. . . . . . . . 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 187	. . . . . . . 8 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
30	29	anbi2i 457	. . . . . . 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 1906	. . . . . . 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 187	. . . . . 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 457	. . . . 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 1906	. . . . 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 187	. . . 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 7596	. . . . 5 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
37		19.42v 1906	. . . . 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 187	. . . 4 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
39	25, 35, 38	3imtr4i 201	. . 3 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) → 𝑞 ∈ (1^st ‘𝐶))
40	39	ex 115	. 2 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) → 𝑞 ∈ (1^st ‘𝐶)))
41	40	rexlimdvw 2598	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 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 {crab 2459 ⟨cop 3595 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 1^st c1st 6138 2^nd c2nd 6139 Qcnq 7278 +_Q cplq 7280 <_Q cltq 7283 Pcnp 7289 <_P cltp 7293

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-eprel 4289 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-oadd 6420 df-omul 6421 df-er 6534 df-ec 6536 df-qs 6540 df-ni 7302 df-pli 7303 df-mi 7304 df-lti 7305 df-plpq 7342 df-enq 7345 df-nqqs 7346 df-plqqs 7347 df-ltnqqs 7351 df-inp 7464 df-iltp 7468

This theorem is referenced by: ltexprlemrnd 7603

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