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Theorem ltexprlemupu 7605

Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7614. (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
ltexprlemupu	⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)) → 𝑟 ∈ (2^nd ‘𝐶)))

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

**Proof of Theorem ltexprlemupu**
Step	Hyp	Ref	Expression
1		simplr 528	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → 𝑟 ∈ Q)
2		simprrr 540	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))
3	2	simpld 112	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → 𝑦 ∈ (1^st ‘𝐴))
4		simprl 529	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → 𝑞 <_Q 𝑟)
5		simpll 527	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → 𝐴<_P 𝐵)
6		simprrl 539	. . . . . . . . . 10 ⊢ ((𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → 𝑦 ∈ (1^st ‘𝐴))
7	6	adantl 277	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → 𝑦 ∈ (1^st ‘𝐴))
8		ltrelpr 7506	. . . . . . . . . . . . 13 ⊢ <_P ⊆ (P × P)
9	8	brel 4680	. . . . . . . . . . . 12 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simpld 112	. . . . . . . . . . 11 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
11		prop 7476	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . . . . 10 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
13		elprnql 7482	. . . . . . . . . 10 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
14	12, 13	sylan 283	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
15	5, 7, 14	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → 𝑦 ∈ Q)
16		ltanqi 7403	. . . . . . . 8 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑦 ∈ Q) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
17	4, 15, 16	syl2anc 411	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
18	9	simprd 114	. . . . . . . . 9 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
19	5, 18	syl 14	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → 𝐵 ∈ P)
20	2	simprd 114	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))
21		prop 7476	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
22		prcunqu 7486	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)) → ((𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟) → (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵)))
23	21, 22	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)) → ((𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟) → (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵)))
24	19, 20, 23	syl2anc 411	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → ((𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟) → (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵)))
25	17, 24	mpd 13	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))
26	1, 3, 25	jca32 310	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → (𝑟 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
27	26	eximi 1600	. . . 4 ⊢ (∃𝑦((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) → ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
28		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))}⟩
29	28	ltexprlemelu 7600	. . . . . . . . 9 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
30		19.42v 1906	. . . . . . . . 9 ⊢ (∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
31	29, 30	bitr4i 187	. . . . . . . 8 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
32	31	anbi2i 457	. . . . . . 7 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ (𝑞 <_Q 𝑟 ∧ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
33		19.42v 1906	. . . . . . 7 ⊢ (∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) ↔ (𝑞 <_Q 𝑟 ∧ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
34	32, 33	bitr4i 187	. . . . . 6 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
35	34	anbi2i 457	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶))) ↔ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))))
36		19.42v 1906	. . . . 5 ⊢ (∃𝑦((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))) ↔ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ ∃𝑦(𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))))
37	35, 36	bitr4i 187	. . . 4 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶))) ↔ ∃𝑦((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))))
38	28	ltexprlemelu 7600	. . . . 5 ⊢ (𝑟 ∈ (2^nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
39		19.42v 1906	. . . . 5 ⊢ (∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
40	38, 39	bitr4i 187	. . . 4 ⊢ (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
41	27, 37, 40	3imtr4i 201	. . 3 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶))) → 𝑟 ∈ (2^nd ‘𝐶))
42	41	ex 115	. 2 ⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) → ((𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)) → 𝑟 ∈ (2^nd ‘𝐶)))
43	42	rexlimdvw 2598	1 ⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)) → 𝑟 ∈ (2^nd ‘𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 {crab 2459 ⟨cop 3597 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 1^st c1st 6141 2^nd c2nd 6142 Qcnq 7281 +_Q cplq 7283 <_Q cltq 7286 Pcnp 7292 <_P cltp 7296

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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-pli 7306 df-mi 7307 df-lti 7308 df-plpq 7345 df-enq 7348 df-nqqs 7349 df-plqqs 7350 df-ltnqqs 7354 df-inp 7467 df-iltp 7471

This theorem is referenced by: ltexprlemrnd 7606

W3C validator