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Theorem ltexprlempr 7888

Description: Our constructed difference is a positive real. Lemma for ltexpri 7893. (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 7880	. . 3 ⊢ (𝐴<_P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐶)))
3		ssrab2 3313	. . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))} ⊆ Q
4		nqex 7643	. . . . . . 7 ⊢ Q ∈ V
5	4	elpw2 4252	. . . . . 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 3313	. . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))} ⊆ Q
8	4	elpw2 4252	. . . . . 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 4763	. . . . 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 2304	. . 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 7885	. . 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 7886	. . 3 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))
16	1	ltexprlemloc 7887	. . 3 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐶) ∨ 𝑟 ∈ (2^nd ‘𝐶))))
17	14, 15, 16	3jca 1204	. 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 7756	. 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 716 ∧ w3a 1005 = wceq 1398 ∃wex 1541 ∈ wcel 2202 ∀wral 2511 ∃wrex 2512 {crab 2515 ⊆ wss 3201 𝒫 cpw 3656 ⟨cop 3676 class class class wbr 4093 × cxp 4729 ‘cfv 5333 (class class class)co 6028 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7560 +_Q cplq 7562 <_Q cltq 7565 Pcnp 7571 <_P cltp 7575

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-1o 6625 df-2o 6626 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7584 df-pli 7585 df-mi 7586 df-lti 7587 df-plpq 7624 df-mpq 7625 df-enq 7627 df-nqqs 7628 df-plqqs 7629 df-mqqs 7630 df-1nqqs 7631 df-rq 7632 df-ltnqqs 7633 df-enq0 7704 df-nq0 7705 df-0nq0 7706 df-plq0 7707 df-mq0 7708 df-inp 7746 df-iltp 7750

This theorem is referenced by: ltexprlemfl 7889 ltexprlemrl 7890 ltexprlemfu 7891 ltexprlemru 7892 ltexpri 7893

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