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Theorem ltexprlempr 6580

Description: Our constructed difference is a positive real. Lemma for ltexpri 6585. (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 6572	. . 3 ⊢ (A<_P B → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐶)))
3		ssrab2 3019	. . . . . 6 ⊢ {x ∈ Q ∣ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q x) ∈ (1^st ‘B))} ⊆ Q
4		nqex 6347	. . . . . . 7 ⊢ Q ∈ V
5	4	elpw2 3902	. . . . . 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 134	. . . . 5 ⊢ {x ∈ Q ∣ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q x) ∈ (1^st ‘B))} ∈ 𝒫 Q
7		ssrab2 3019	. . . . . 6 ⊢ {x ∈ Q ∣ ∃y(y ∈ (1^st ‘A) ∧ (y +_Q x) ∈ (2^nd ‘B))} ⊆ Q
8	4	elpw2 3902	. . . . . 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 134	. . . . 5 ⊢ {x ∈ Q ∣ ∃y(y ∈ (1^st ‘A) ∧ (y +_Q x) ∈ (2^nd ‘B))} ∈ 𝒫 Q
10		opelxpi 4319	. . . . 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 402	. . . 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 2107	. . 3 ⊢ 𝐶 ∈ (𝒫 Q × 𝒫 Q)
13	2, 12	jctil 295	. 2 ⊢ (A<_P B → (𝐶 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐶))))
14	1	ltexprlemrnd 6577	. . 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 6578	. . 3 ⊢ (A<_P B → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))
16	1	ltexprlemloc 6579	. . 3 ⊢ (A<_P B → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘𝐶) ∨ 𝑟 ∈ (2^nd ‘𝐶))))
17	14, 15, 16	3jca 1083	. 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 6458	. 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 394	1 ⊢ (A<_P B → 𝐶 ∈ 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 ∀wral 2300 ∃wrex 2301 {crab 2304 ⊆ wss 2911 𝒫 cpw 3351 ⟨cop 3370 class class class wbr 3755 × cxp 4286 ‘cfv 4845 (class class class)co 5455 1^st c1st 5707 2^nd c2nd 5708 Qcnq 6264 +_Q cplq 6266 <_Q cltq 6269 Pcnp 6275 <_P cltp 6279

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-bnd 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-2o 5941 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-enq0 6406 df-nq0 6407 df-0nq0 6408 df-plq0 6409 df-mq0 6410 df-inp 6448 df-iltp 6452

This theorem is referenced by: ltexprlemfl 6581 ltexprlemrl 6582 ltexprlemfu 6583 ltexprlemru 6584 ltexpri 6585

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