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Mirrors > Home > ILE Home > Th. List > ltnqpri | GIF version |
Description: We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.) |
Ref | Expression |
---|---|
ltnqpri | ⊢ (𝐴 <Q 𝐵 → 〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 6463 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4392 | . . . . . . 7 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q)) |
3 | 2 | simpld 105 | . . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐴 ∈ Q) |
4 | nqprlu 6645 | . . . . . 6 ⊢ (𝐴 ∈ Q → 〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 ∈ P) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝐴 <Q 𝐵 → 〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 ∈ P) |
6 | 2 | simprd 107 | . . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐵 ∈ Q) |
7 | nqprlu 6645 | . . . . . 6 ⊢ (𝐵 ∈ Q → 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉 ∈ P) | |
8 | 6, 7 | syl 14 | . . . . 5 ⊢ (𝐴 <Q 𝐵 → 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉 ∈ P) |
9 | ltdfpr 6604 | . . . . 5 ⊢ ((〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉 ∈ P ∧ 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉 ∈ P) → (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ∧ 𝑥 ∈ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉)))) | |
10 | 5, 8, 9 | syl2anc 391 | . . . 4 ⊢ (𝐴 <Q 𝐵 → (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ∧ 𝑥 ∈ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉)))) |
11 | vex 2560 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
12 | breq2 3768 | . . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑥)) | |
13 | ltnqex 6647 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V | |
14 | gtnqex 6648 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V | |
15 | 13, 14 | op2nd 5774 | . . . . . . 7 ⊢ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) = {𝑢 ∣ 𝐴 <Q 𝑢} |
16 | 11, 12, 15 | elab2 2690 | . . . . . 6 ⊢ (𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ↔ 𝐴 <Q 𝑥) |
17 | breq1 3767 | . . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐵 ↔ 𝑥 <Q 𝐵)) | |
18 | ltnqex 6647 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V | |
19 | gtnqex 6648 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V | |
20 | 18, 19 | op1st 5773 | . . . . . . 7 ⊢ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) = {𝑙 ∣ 𝑙 <Q 𝐵} |
21 | 11, 17, 20 | elab2 2690 | . . . . . 6 ⊢ (𝑥 ∈ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) ↔ 𝑥 <Q 𝐵) |
22 | 16, 21 | anbi12i 433 | . . . . 5 ⊢ ((𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ∧ 𝑥 ∈ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉)) ↔ (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)) |
23 | 22 | rexbii 2331 | . . . 4 ⊢ (∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ∧ 𝑥 ∈ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉)) ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)) |
24 | 10, 23 | syl6bb 185 | . . 3 ⊢ (𝐴 <Q 𝐵 → (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉 ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵))) |
25 | ltbtwnnqq 6513 | . . 3 ⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)) | |
26 | 24, 25 | syl6bbr 187 | . 2 ⊢ (𝐴 <Q 𝐵 → (〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉 ↔ 𝐴 <Q 𝐵)) |
27 | 26 | ibir 166 | 1 ⊢ (𝐴 <Q 𝐵 → 〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1393 {cab 2026 ∃wrex 2307 〈cop 3378 class class class wbr 3764 ‘cfv 4902 1st c1st 5765 2nd c2nd 5766 Qcnq 6378 <Q cltq 6383 Pcnp 6389 <P cltp 6393 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 df-iltp 6568 |
This theorem is referenced by: caucvgprprlemk 6781 caucvgprprlemloccalc 6782 caucvgprprlemnjltk 6789 caucvgprprlemlol 6796 caucvgprprlemupu 6798 |
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