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Mirrors > Home > ILE Home > Th. List > ltexprlempr | GIF version |
Description: Our constructed difference is a positive real. Lemma for ltexpri 6587. (Contributed by Jim Kingdon, 17-Dec-2019.) |
Ref | Expression |
---|---|
ltexprlem.1 | ⊢ 𝐶 = 〈{x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))}〉 |
Ref | Expression |
---|---|
ltexprlempr | ⊢ (A<P B → 𝐶 ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltexprlem.1 | . . . 4 ⊢ 𝐶 = 〈{x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))}〉 | |
2 | 1 | ltexprlemm 6574 | . . 3 ⊢ (A<P B → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))) |
3 | ssrab2 3019 | . . . . . 6 ⊢ {x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))} ⊆ Q | |
4 | nqex 6347 | . . . . . . 7 ⊢ Q ∈ V | |
5 | 4 | elpw2 3902 | . . . . . 6 ⊢ ({x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))} ∈ 𝒫 Q ↔ {x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))} ⊆ Q) |
6 | 3, 5 | mpbir 134 | . . . . 5 ⊢ {x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))} ∈ 𝒫 Q |
7 | ssrab2 3019 | . . . . . 6 ⊢ {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))} ⊆ Q | |
8 | 4 | elpw2 3902 | . . . . . 6 ⊢ ({x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))} ∈ 𝒫 Q ↔ {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))} ⊆ Q) |
9 | 7, 8 | mpbir 134 | . . . . 5 ⊢ {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))} ∈ 𝒫 Q |
10 | opelxpi 4319 | . . . . 5 ⊢ (({x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))} ∈ 𝒫 Q ∧ {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))} ∈ 𝒫 Q) → 〈{x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘B))}〉 ∈ (𝒫 Q × 𝒫 Q)) | |
11 | 6, 9, 10 | mp2an 402 | . . . 4 ⊢ 〈{x ∈ Q ∣ ∃y(y ∈ (2nd ‘A) ∧ (y +Q x) ∈ (1st ‘B))}, {x ∈ Q ∣ ∃y(y ∈ (1st ‘A) ∧ (y +Q x) ∈ (2nd ‘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 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))) |
14 | 1 | ltexprlemrnd 6579 | . . 3 ⊢ (A<P B → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶))))) |
15 | 1 | ltexprlemdisj 6580 | . . 3 ⊢ (A<P B → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶))) |
16 | 1 | ltexprlemloc 6581 | . . 3 ⊢ (A<P B → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)))) |
17 | 14, 15, 16 | 3jca 1083 | . 2 ⊢ (A<P B → ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))) |
18 | elnp1st2nd 6459 | . 2 ⊢ (𝐶 ∈ P ↔ ((𝐶 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘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 1st c1st 5707 2nd 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-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-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 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-iltp 6453 |
This theorem is referenced by: ltexprlemfl 6583 ltexprlemrl 6584 ltexprlemfu 6585 ltexprlemru 6586 ltexpri 6587 |
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