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Theorem ltexprlempr 7818

Description: Our constructed difference is a positive real. Lemma for ltexpri 7823. (Contributed by Jim Kingdon, 17-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
ltexprlempr	⊢ (𝐴<_P 𝐵 → 𝐶 ∈ P)

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

Proof of Theorem ltexprlempr Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . 4 ⊢ 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))}⟩
2	1	ltexprlemm 7810	. . 3 ⊢ (𝐴<_P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐶)))
3		ssrab2 3310	. . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ⊆ Q
4		nqex 7573	. . . . . . 7 ⊢ Q ∈ V
5	4	elpw2 4245	. . . . . 6 ⊢ ({𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ∈ 𝒫 Q ↔ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ⊆ Q)
6	3, 5	mpbir 146	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ∈ 𝒫 Q
7		ssrab2 3310	. . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ⊆ Q
8	4	elpw2 4245	. . . . . 6 ⊢ ({𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ∈ 𝒫 Q ↔ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ⊆ Q)
9	7, 8	mpbir 146	. . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ∈ 𝒫 Q
10		opelxpi 4755	. . . . 5 ⊢ (({𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ∈ 𝒫 Q ∧ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ∈ 𝒫 Q) → ⟨{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))}⟩ ∈ (𝒫 Q × 𝒫 Q))
11	6, 9, 10	mp2an 426	. . . 4 ⊢ ⟨{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))}⟩ ∈ (𝒫 Q × 𝒫 Q)
12	1, 11	eqeltri 2302	. . 3 ⊢ 𝐶 ∈ (𝒫 Q × 𝒫 Q)
13	2, 12	jctil 312	. 2 ⊢ (𝐴<_P 𝐵 → (𝐶 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐶))))
14	1	ltexprlemrnd 7815	. . 3 ⊢ (𝐴<_P 𝐵 → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)))))
15	1	ltexprlemdisj 7816	. . 3 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))
16	1	ltexprlemloc 7817	. . 3 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐶) ∨ 𝑟 ∈ (2^nd ‘𝐶))))
17	14, 15, 16	3jca 1201	. 2 ⊢ (𝐴<_P 𝐵 → ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐶) ∨ 𝑟 ∈ (2^nd ‘𝐶)))))
18		elnp1st2nd 7686	. 2 ⊢ (𝐶 ∈ P ↔ ((𝐶 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐶))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐶) ∨ 𝑟 ∈ (2^nd ‘𝐶))))))
19	13, 17, 18	sylanbrc 417	1 ⊢ (𝐴<_P 𝐵 → 𝐶 ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 713 ∧ w3a 1002 = wceq 1395 ∃wex 1538 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 {crab 2512 ⊆ wss 3198 𝒫 cpw 3650 ⟨cop 3670 class class class wbr 4086 × cxp 4721 ‘cfv 5324 (class class class)co 6013 1^st c1st 6296 2^nd c2nd 6297 Qcnq 7490 +_Q cplq 7492 <_Q cltq 7495 Pcnp 7501 <_P cltp 7505

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-eprel 4384 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-1o 6577 df-2o 6578 df-oadd 6581 df-omul 6582 df-er 6697 df-ec 6699 df-qs 6703 df-ni 7514 df-pli 7515 df-mi 7516 df-lti 7517 df-plpq 7554 df-mpq 7555 df-enq 7557 df-nqqs 7558 df-plqqs 7559 df-mqqs 7560 df-1nqqs 7561 df-rq 7562 df-ltnqqs 7563 df-enq0 7634 df-nq0 7635 df-0nq0 7636 df-plq0 7637 df-mq0 7638 df-inp 7676 df-iltp 7680

This theorem is referenced by: ltexprlemfl 7819 ltexprlemrl 7820 ltexprlemfu 7821 ltexprlemru 7822 ltexpri 7823

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