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Theorem ltexprlemlol 7865

Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7876. (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 529	. . . . . 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 542	. . . . . . 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 531	. . . . . . . 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 7768	. . . . . . . . . . . 12 ⊢ <_P ⊆ (P × P)
7	6	brel 4784	. . . . . . . . . . 11 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
8	7	simpld 112	. . . . . . . . . 10 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
9		prop 7738	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
10		elprnqu 7745	. . . . . . . . . . 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 7665	. . . . . . . 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 7738	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
20		prcdnql 7747	. . . . . . . . 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 1649	. . . 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 7861	. . . . . . . . 9 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (1^st ‘𝐵))))
28		19.42v 1955	. . . . . . . . 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 1955	. . . . . . 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 1955	. . . . 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 7861	. . . . 5 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
37		19.42v 1955	. . . . 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 2655	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 1398 ∃wex 1541 ∈ wcel 2202 ∃wrex 2512 {crab 2515 ⟨cop 3676 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7543 +_Q cplq 7545 <_Q cltq 7548 Pcnp 7554 <_P cltp 7558

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7567 df-pli 7568 df-mi 7569 df-lti 7570 df-plpq 7607 df-enq 7610 df-nqqs 7611 df-plqqs 7612 df-ltnqqs 7616 df-inp 7729 df-iltp 7733

This theorem is referenced by: ltexprlemrnd 7868

W3C validator