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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 6924 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 4490 | . . . . . . 7 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q)) |
3 | 2 | simpld 110 | . . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐴 ∈ Q) |
4 | nqprlu 7106 | . . . . . 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 7106 | . . . . . 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 7065 | . . . . 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 2622 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
12 | breq2 3849 | . . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑥)) | |
13 | ltnqex 7108 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V | |
14 | gtnqex 7109 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V | |
15 | 13, 14 | op2nd 5918 | . . . . . . 7 ⊢ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) = {𝑢 ∣ 𝐴 <Q 𝑢} |
16 | 11, 12, 15 | elab2 2763 | . . . . . 6 ⊢ (𝑥 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉) ↔ 𝐴 <Q 𝑥) |
17 | breq1 3848 | . . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐵 ↔ 𝑥 <Q 𝐵)) | |
18 | ltnqex 7108 | . . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V | |
19 | gtnqex 7109 | . . . . . . . 8 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V | |
20 | 18, 19 | op1st 5917 | . . . . . . 7 ⊢ (1st ‘〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) = {𝑙 ∣ 𝑙 <Q 𝐵} |
21 | 11, 17, 20 | elab2 2763 | . . . . . 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 2385 | . . . 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 6974 | . . 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 1438 {cab 2074 ∃wrex 2360 〈cop 3449 class class class wbr 3845 ‘cfv 5015 1st c1st 5909 2nd c2nd 5910 Qcnq 6839 <Q cltq 6844 Pcnp 6850 <P cltp 6854 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-eprel 4116 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-1o 6181 df-oadd 6185 df-omul 6186 df-er 6292 df-ec 6294 df-qs 6298 df-ni 6863 df-pli 6864 df-mi 6865 df-lti 6866 df-plpq 6903 df-mpq 6904 df-enq 6906 df-nqqs 6907 df-plqqs 6908 df-mqqs 6909 df-1nqqs 6910 df-rq 6911 df-ltnqqs 6912 df-inp 7025 df-iltp 7029 |
This theorem is referenced by: caucvgprprlemk 7242 caucvgprprlemloccalc 7243 caucvgprprlemnjltk 7250 caucvgprprlemlol 7257 caucvgprprlemupu 7259 |
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