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Theorem recexprlempr 7989

Description: 𝐵 is a positive real. Lemma for recexpr 7995. (Contributed by Jim Kingdon, 27-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlempr	⊢ (𝐴 ∈ P → 𝐵 ∈ P)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦

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

Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemm 7981	. . 3 ⊢ (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵)))
3		ltrelnq 7722	. . . . . . . . . . 11 ⊢ <_Q ⊆ (Q × Q)
4	3	brel 4822	. . . . . . . . . 10 ⊢ (𝑥 <_Q 𝑦 → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
5	4	simpld 112	. . . . . . . . 9 ⊢ (𝑥 <_Q 𝑦 → 𝑥 ∈ Q)
6	5	adantr 276	. . . . . . . 8 ⊢ ((𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
7	6	exlimiv 1651	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
8	7	abssi 3323	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
9		nqex 7720	. . . . . . 7 ⊢ Q ∈ V
10	9	elpw2 4288	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ 𝒫 Q ↔ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q)
11	8, 10	mpbir 146	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ 𝒫 Q
12	3	brel 4822	. . . . . . . . . 10 ⊢ (𝑦 <_Q 𝑥 → (𝑦 ∈ Q ∧ 𝑥 ∈ Q))
13	12	simprd 114	. . . . . . . . 9 ⊢ (𝑦 <_Q 𝑥 → 𝑥 ∈ Q)
14	13	adantr 276	. . . . . . . 8 ⊢ ((𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
15	14	exlimiv 1651	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
16	15	abssi 3323	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
17	9	elpw2 4288	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ 𝒫 Q ↔ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q)
18	16, 17	mpbir 146	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ 𝒫 Q
19		opelxpi 4801	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ 𝒫 Q ∧ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ 𝒫 Q) → ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩ ∈ (𝒫 Q × 𝒫 Q))
20	11, 18, 19	mp2an 430	. . . 4 ⊢ ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩ ∈ (𝒫 Q × 𝒫 Q)
21	1, 20	eqeltri 2311	. . 3 ⊢ 𝐵 ∈ (𝒫 Q × 𝒫 Q)
22	2, 21	jctil 312	. 2 ⊢ (𝐴 ∈ P → (𝐵 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵))))
23	1	recexprlemrnd 7986	. . 3 ⊢ (𝐴 ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))))
24	1	recexprlemdisj 7987	. . 3 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐵) ∧ 𝑞 ∈ (2^nd ‘𝐵)))
25	1	recexprlemloc 7988	. . 3 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐵) ∨ 𝑟 ∈ (2^nd ‘𝐵))))
26	23, 24, 25	3jca 1208	. 2 ⊢ (𝐴 ∈ P → ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐵) ∧ 𝑞 ∈ (2^nd ‘𝐵)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐵) ∨ 𝑟 ∈ (2^nd ‘𝐵)))))
27		elnp1st2nd 7833	. 2 ⊢ (𝐵 ∈ P ↔ ((𝐵 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐵) ∧ 𝑞 ∈ (2^nd ‘𝐵)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐵) ∨ 𝑟 ∈ (2^nd ‘𝐵))))))
28	22, 26, 27	sylanbrc 421	1 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 {cab 2224 ∀wral 2528 ∃wrex 2529 ⊆ wss 3220 𝒫 cpw 3685 ⟨cop 3708 class class class wbr 4125 × cxp 4767 ‘cfv 5372 1^st c1st 6362 2^nd c2nd 6363 Qcnq 7637 *_Qcrq 7641 <_Q cltq 7642 Pcnp 7648

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-eprel 4429 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-1o 6677 df-oadd 6681 df-omul 6682 df-er 6797 df-ec 6799 df-qs 6803 df-ni 7661 df-pli 7662 df-mi 7663 df-lti 7664 df-plpq 7701 df-mpq 7702 df-enq 7704 df-nqqs 7705 df-plqqs 7706 df-mqqs 7707 df-1nqqs 7708 df-rq 7709 df-ltnqqs 7710 df-inp 7823

This theorem is referenced by: recexprlem1ssl 7990 recexprlem1ssu 7991 recexprlemss1l 7992 recexprlemss1u 7993 recexprlemex 7994 recexpr 7995

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