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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 7173 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4591 | . . . . . . 7 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q)) |
3 | 2 | simpld 111 | . . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐴 ∈ Q) |
4 | nqprlu 7355 | . . . . . 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 113 | . . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐵 ∈ Q) |
7 | nqprlu 7355 | . . . . . 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 7314 | . . . . 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 408 | . . . 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 2689 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
12 | breq2 3933 | . . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑥)) | |
13 | ltnqex 7357 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V | |
14 | gtnqex 7358 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V | |
15 | 13, 14 | op2nd 6045 | . . . . . . 7 ⊢ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) = {𝑢 ∣ 𝐴 <Q 𝑢} |
16 | 11, 12, 15 | elab2 2832 | . . . . . 6 ⊢ (𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ↔ 𝐴 <Q 𝑥) |
17 | breq1 3932 | . . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐵 ↔ 𝑥 <Q 𝐵)) | |
18 | ltnqex 7357 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V | |
19 | gtnqex 7358 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V | |
20 | 18, 19 | op1st 6044 | . . . . . . 7 ⊢ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) = {𝑙 ∣ 𝑙 <Q 𝐵} |
21 | 11, 17, 20 | elab2 2832 | . . . . . 6 ⊢ (𝑥 ∈ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) ↔ 𝑥 <Q 𝐵) |
22 | 16, 21 | anbi12i 455 | . . . . 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 2442 | . . . 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 195 | . . 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 7223 | . . 3 ⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)) | |
26 | 24, 25 | syl6bbr 197 | . 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 176 | 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 103 ↔ wb 104 ∈ wcel 1480 {cab 2125 ∃wrex 2417 〈cop 3530 class class class wbr 3929 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 Qcnq 7088 <Q cltq 7093 Pcnp 7099 <P cltp 7103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-inp 7274 df-iltp 7278 |
This theorem is referenced by: caucvgprprlemk 7491 caucvgprprlemloccalc 7492 caucvgprprlemnjltk 7499 caucvgprprlemlol 7506 caucvgprprlemupu 7508 suplocexprlemloc 7529 |
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