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Theorem ltexprlempr 7795

Description: Our constructed difference is a positive real. Lemma for ltexpri 7800. (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 7787	. . 3 ⊢ (𝐴<_P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐶)))
3		ssrab2 3309	. . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ⊆ Q
4		nqex 7550	. . . . . . 7 ⊢ Q ∈ V
5	4	elpw2 4241	. . . . . 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 3309	. . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ⊆ Q
8	4	elpw2 4241	. . . . . 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 4751	. . . . 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 7792	. . 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 7793	. . 3 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))
16	1	ltexprlemloc 7794	. . 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 7663	. 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 3197 𝒫 cpw 3649 ⟨cop 3669 class class class wbr 4083 × cxp 4717 ‘cfv 5318 (class class class)co 6001 1^st c1st 6284 2^nd c2nd 6285 Qcnq 7467 +_Q cplq 7469 <_Q cltq 7472 Pcnp 7478 <_P cltp 7482

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 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

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 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-eprel 4380 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-1o 6562 df-2o 6563 df-oadd 6566 df-omul 6567 df-er 6680 df-ec 6682 df-qs 6686 df-ni 7491 df-pli 7492 df-mi 7493 df-lti 7494 df-plpq 7531 df-mpq 7532 df-enq 7534 df-nqqs 7535 df-plqqs 7536 df-mqqs 7537 df-1nqqs 7538 df-rq 7539 df-ltnqqs 7540 df-enq0 7611 df-nq0 7612 df-0nq0 7613 df-plq0 7614 df-mq0 7615 df-inp 7653 df-iltp 7657

This theorem is referenced by: ltexprlemfl 7796 ltexprlemrl 7797 ltexprlemfu 7798 ltexprlemru 7799 ltexpri 7800

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