![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ltexprlempr | GIF version |
Description: Our constructed difference is a positive real. Lemma for ltexpri 7643. (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 7630 | . . 3 ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))) |
3 | ssrab2 3255 | . . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))} ⊆ Q | |
4 | nqex 7393 | . . . . . . 7 ⊢ Q ∈ V | |
5 | 4 | elpw2 4175 | . . . . . 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 146 | . . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))} ∈ 𝒫 Q |
7 | ssrab2 3255 | . . . . . 6 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))} ⊆ Q | |
8 | 4 | elpw2 4175 | . . . . . 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 146 | . . . . 5 ⊢ {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))} ∈ 𝒫 Q |
10 | opelxpi 4676 | . . . . 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 426 | . . . 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 2262 | . . 3 ⊢ 𝐶 ∈ (𝒫 Q × 𝒫 Q) |
13 | 2, 12 | jctil 312 | . 2 ⊢ (𝐴<P 𝐵 → (𝐶 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))) |
14 | 1 | ltexprlemrnd 7635 | . . 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 7636 | . . 3 ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶))) |
16 | 1 | ltexprlemloc 7637 | . . 3 ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)))) |
17 | 14, 15, 16 | 3jca 1179 | . 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 7506 | . 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 417 | 1 ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ∀wral 2468 ∃wrex 2469 {crab 2472 ⊆ wss 3144 𝒫 cpw 3590 〈cop 3610 class class class wbr 4018 × cxp 4642 ‘cfv 5235 (class class class)co 5897 1st c1st 6164 2nd c2nd 6165 Qcnq 7310 +Q cplq 7312 <Q cltq 7315 Pcnp 7321 <P cltp 7325 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-1o 6442 df-2o 6443 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-pli 7335 df-mi 7336 df-lti 7337 df-plpq 7374 df-mpq 7375 df-enq 7377 df-nqqs 7378 df-plqqs 7379 df-mqqs 7380 df-1nqqs 7381 df-rq 7382 df-ltnqqs 7383 df-enq0 7454 df-nq0 7455 df-0nq0 7456 df-plq0 7457 df-mq0 7458 df-inp 7496 df-iltp 7500 |
This theorem is referenced by: ltexprlemfl 7639 ltexprlemrl 7640 ltexprlemfu 7641 ltexprlemru 7642 ltexpri 7643 |
Copyright terms: Public domain | W3C validator |