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Theorem ltexprlemupu 7405

Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7414. (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 519	. . . . . 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 529	. . . . . . 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 111	. . . . . 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 520	. . . . . . . 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 518	. . . . . . . . 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 528	. . . . . . . . . 10 ⊢ ((𝑞 <_Q 𝑟 ∧ (𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → 𝑦 ∈ (1^st ‘𝐴))
7	6	adantl 275	. . . . . . . . 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 7306	. . . . . . . . . . . . 13 ⊢ <_P ⊆ (P × P)
9	8	brel 4586	. . . . . . . . . . . 12 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
10	9	simpld 111	. . . . . . . . . . 11 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
11		prop 7276	. . . . . . . . . . 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 7282	. . . . . . . . . 10 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
14	12, 13	sylan 281	. . . . . . . . 9 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
15	5, 7, 14	syl2anc 408	. . . . . . . 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 7203	. . . . . . . 8 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑦 ∈ Q) → (𝑦 +_Q 𝑞) <_Q (𝑦 +_Q 𝑟))
17	4, 15, 16	syl2anc 408	. . . . . . 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 113	. . . . . . . . 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 113	. . . . . . . 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 7276	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
22		prcunqu 7286	. . . . . . . . 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 281	. . . . . . . 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 408	. . . . . . 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 308	. . . . 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 1579	. . . 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 7400	. . . . . . . . 9 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
30		19.42v 1878	. . . . . . . . 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 186	. . . . . . . 8 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ ∃𝑦(𝑞 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
32	31	anbi2i 452	. . . . . . 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 1878	. . . . . . 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 186	. . . . . 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 452	. . . . 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 1878	. . . . 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 186	. . . 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 7400	. . . . 5 ⊢ (𝑟 ∈ (2^nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
39		19.42v 1878	. . . . 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 186	. . . 4 ⊢ (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑦(𝑟 ∈ Q ∧ (𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑟) ∈ (2^nd ‘𝐵))))
41	27, 37, 40	3imtr4i 200	. . 3 ⊢ (((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) ∧ (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶))) → 𝑟 ∈ (2^nd ‘𝐶))
42	41	ex 114	. 2 ⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) → ((𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)) → 𝑟 ∈ (2^nd ‘𝐶)))
43	42	rexlimdvw 2551	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 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∃wrex 2415 {crab 2418 ⟨cop 3525 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 1^st c1st 6029 2^nd c2nd 6030 Qcnq 7081 +_Q cplq 7083 <_Q cltq 7086 Pcnp 7092 <_P cltp 7096

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-ltnqqs 7154 df-inp 7267 df-iltp 7271

This theorem is referenced by: ltexprlemrnd 7406

W3C validator