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Theorem recexprlempr 6604

Description: B is a positive real. Lemma for recexpr 6610. (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 6596	. . 3 ⊢ (A ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘B) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘B)))
3		ltrelnq 6349	. . . . . . . . . . 11 ⊢ <_Q ⊆ (Q × Q)
4	3	brel 4335	. . . . . . . . . 10 ⊢ (x <_Q y → (x ∈ Q ∧ y ∈ Q))
5	4	simpld 105	. . . . . . . . 9 ⊢ (x <_Q y → x ∈ Q)
6	5	adantr 261	. . . . . . . 8 ⊢ ((x <_Q y ∧ (*_Q‘y) ∈ (2^nd ‘A)) → x ∈ Q)
7	6	exlimiv 1486	. . . . . . 7 ⊢ (∃y(x <_Q y ∧ (*_Q‘y) ∈ (2^nd ‘A)) → x ∈ Q)
8	7	abssi 3009	. . . . . 6 ⊢ {x ∣ ∃y(x <_Q y ∧ (*_Q‘y) ∈ (2^nd ‘A))} ⊆ Q
9		nqex 6347	. . . . . . 7 ⊢ Q ∈ V
10	9	elpw2 3902	. . . . . 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 134	. . . . 5 ⊢ {x ∣ ∃y(x <_Q y ∧ (*_Q‘y) ∈ (2^nd ‘A))} ∈ 𝒫 Q
12	3	brel 4335	. . . . . . . . . 10 ⊢ (y <_Q x → (y ∈ Q ∧ x ∈ Q))
13	12	simprd 107	. . . . . . . . 9 ⊢ (y <_Q x → x ∈ Q)
14	13	adantr 261	. . . . . . . 8 ⊢ ((y <_Q x ∧ (*_Q‘y) ∈ (1^st ‘A)) → x ∈ Q)
15	14	exlimiv 1486	. . . . . . 7 ⊢ (∃y(y <_Q x ∧ (*_Q‘y) ∈ (1^st ‘A)) → x ∈ Q)
16	15	abssi 3009	. . . . . 6 ⊢ {x ∣ ∃y(y <_Q x ∧ (*_Q‘y) ∈ (1^st ‘A))} ⊆ Q
17	9	elpw2 3902	. . . . . 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 134	. . . . 5 ⊢ {x ∣ ∃y(y <_Q x ∧ (*_Q‘y) ∈ (1^st ‘A))} ∈ 𝒫 Q
19		opelxpi 4319	. . . . 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 402	. . . 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 2107	. . 3 ⊢ B ∈ (𝒫 Q × 𝒫 Q)
22	2, 21	jctil 295	. 2 ⊢ (A ∈ P → (B ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘B) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘B))))
23	1	recexprlemrnd 6601	. . 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 6602	. . 3 ⊢ (A ∈ P → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘B) ∧ 𝑞 ∈ (2^nd ‘B)))
25	1	recexprlemloc 6603	. . 3 ⊢ (A ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘B) ∨ 𝑟 ∈ (2^nd ‘B))))
26	23, 24, 25	3jca 1083	. 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 6459	. 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 394	1 ⊢ (A ∈ P → B ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 ∧ w3a 884 = wceq 1242 ∃wex 1378 ∈ wcel 1390 {cab 2023 ∀wral 2300 ∃wrex 2301 ⊆ wss 2911 𝒫 cpw 3351 ⟨cop 3370 class class class wbr 3755 × cxp 4286 ‘cfv 4845 1^st c1st 5707 2^nd c2nd 5708 Qcnq 6264 *_Qcrq 6268 <_Q cltq 6269 Pcnp 6275

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-inp 6449

This theorem is referenced by: recexprlem1ssl 6605 recexprlem1ssu 6606 recexprlemss1l 6607 recexprlemss1u 6608 recexprlemex 6609 recexpr 6610

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