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Theorem ltexprlemlol 6574

Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 6585. (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 6487	. . . . . . . . . . . 12 ⊢ <_P ⊆ (P × P)
7	6	brel 4335	. . . . . . . . . . 11 ⊢ (A<_P B → (A ∈ P ∧ B ∈ P))
8	7	simpld 105	. . . . . . . . . 10 ⊢ (A<_P B → A ∈ P)
9		prop 6457	. . . . . . . . . . 11 ⊢ (A ∈ P → ⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P)
10		elprnqu 6464	. . . . . . . . . . 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 6386	. . . . . . . 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 6457	. . . . . . . . 9 ⊢ (B ∈ P → ⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P)
20		prcdnql 6466	. . . . . . . . 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 1488	. . . 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 6570	. . . . . . . . 9 ⊢ (𝑟 ∈ (1^st ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q 𝑟) ∈ (1^st ‘B))))
28		19.42v 1783	. . . . . . . . 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 1783	. . . . . . 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 1783	. . . . 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 6570	. . . . 5 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q 𝑞) ∈ (1^st ‘B))))
37		19.42v 1783	. . . . 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 2430	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 1242 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 {crab 2304 ⟨cop 3370 class class class wbr 3755 ‘cfv 4845 (class class class)co 5455 1^st c1st 5707 2^nd c2nd 5708 Qcnq 6264 +_Q cplq 6266 <_Q cltq 6269 Pcnp 6275 <_P cltp 6279

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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bnd 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-ltnqqs 6337 df-inp 6448 df-iltp 6452

This theorem is referenced by: ltexprlemrnd 6577

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