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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 6669 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4438 | . . . . . . 7 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q)) |
3 | 2 | simpld 110 | . . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐴 ∈ Q) |
4 | nqprlu 6851 | . . . . . 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 112 | . . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐵 ∈ Q) |
7 | nqprlu 6851 | . . . . . 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 6810 | . . . . 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 403 | . . . 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 2613 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
12 | breq2 3809 | . . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑥)) | |
13 | ltnqex 6853 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V | |
14 | gtnqex 6854 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V | |
15 | 13, 14 | op2nd 5825 | . . . . . . 7 ⊢ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) = {𝑢 ∣ 𝐴 <Q 𝑢} |
16 | 11, 12, 15 | elab2 2749 | . . . . . 6 ⊢ (𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ↔ 𝐴 <Q 𝑥) |
17 | breq1 3808 | . . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐵 ↔ 𝑥 <Q 𝐵)) | |
18 | ltnqex 6853 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V | |
19 | gtnqex 6854 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V | |
20 | 18, 19 | op1st 5824 | . . . . . . 7 ⊢ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) = {𝑙 ∣ 𝑙 <Q 𝐵} |
21 | 11, 17, 20 | elab2 2749 | . . . . . 6 ⊢ (𝑥 ∈ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) ↔ 𝑥 <Q 𝐵) |
22 | 16, 21 | anbi12i 448 | . . . . 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 2378 | . . . 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 194 | . . 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 6719 | . . 3 ⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)) | |
26 | 24, 25 | syl6bbr 196 | . 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 175 | 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 102 ↔ wb 103 ∈ wcel 1434 {cab 2069 ∃wrex 2354 〈cop 3419 class class class wbr 3805 ‘cfv 4952 1st c1st 5816 2nd c2nd 5817 Qcnq 6584 <Q cltq 6589 Pcnp 6595 <P cltp 6599 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-eprel 4072 df-id 4076 df-po 4079 df-iso 4080 df-iord 4149 df-on 4151 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-1st 5818 df-2nd 5819 df-recs 5974 df-irdg 6039 df-1o 6085 df-oadd 6089 df-omul 6090 df-er 6193 df-ec 6195 df-qs 6199 df-ni 6608 df-pli 6609 df-mi 6610 df-lti 6611 df-plpq 6648 df-mpq 6649 df-enq 6651 df-nqqs 6652 df-plqqs 6653 df-mqqs 6654 df-1nqqs 6655 df-rq 6656 df-ltnqqs 6657 df-inp 6770 df-iltp 6774 |
This theorem is referenced by: caucvgprprlemk 6987 caucvgprprlemloccalc 6988 caucvgprprlemnjltk 6995 caucvgprprlemlol 7002 caucvgprprlemupu 7004 |
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