Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ltposr | GIF version |
Description: Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Ref | Expression |
---|---|
ltposr | ⊢ <R Po R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7535 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
2 | id 19 | . . . . . . 7 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝑓 → [〈𝑥, 𝑦〉] ~R = 𝑓) | |
3 | 2, 2 | breq12d 3942 | . . . . . 6 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝑓 → ([〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ↔ 𝑓 <R 𝑓)) |
4 | 3 | notbid 656 | . . . . 5 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝑓 → (¬ [〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ↔ ¬ 𝑓 <R 𝑓)) |
5 | ltsopr 7404 | . . . . . . . 8 ⊢ <P Or P | |
6 | ltrelpr 7313 | . . . . . . . 8 ⊢ <P ⊆ (P × P) | |
7 | 5, 6 | soirri 4933 | . . . . . . 7 ⊢ ¬ (𝑥 +P 𝑦)<P (𝑥 +P 𝑦) |
8 | addcomprg 7386 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥)) | |
9 | 8 | breq2d 3941 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 𝑦)<P (𝑥 +P 𝑦) ↔ (𝑥 +P 𝑦)<P (𝑦 +P 𝑥))) |
10 | 7, 9 | mtbii 663 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ¬ (𝑥 +P 𝑦)<P (𝑦 +P 𝑥)) |
11 | ltsrprg 7555 | . . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ↔ (𝑥 +P 𝑦)<P (𝑦 +P 𝑥))) | |
12 | 11 | anidms 394 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ↔ (𝑥 +P 𝑦)<P (𝑦 +P 𝑥))) |
13 | 10, 12 | mtbird 662 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ¬ [〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ) |
14 | 1, 4, 13 | ecoptocl 6516 | . . . 4 ⊢ (𝑓 ∈ R → ¬ 𝑓 <R 𝑓) |
15 | 14 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑓 ∈ R) → ¬ 𝑓 <R 𝑓) |
16 | lttrsr 7570 | . . . 4 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ)) | |
17 | 16 | adantl 275 | . . 3 ⊢ ((⊤ ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → ((𝑓 <R 𝑔 ∧ 𝑔 <R ℎ) → 𝑓 <R ℎ)) |
18 | 15, 17 | ispod 4226 | . 2 ⊢ (⊤ → <R Po R) |
19 | 18 | mptru 1340 | 1 ⊢ <R Po R |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 Po wpo 4216 (class class class)co 5774 [cec 6427 Pcnp 7099 +P cpp 7101 <P cltp 7103 ~R cer 7104 Rcnr 7105 <R cltr 7111 |
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-2o 6314 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-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-iplp 7276 df-iltp 7278 df-enr 7534 df-nr 7535 df-ltr 7538 |
This theorem is referenced by: ltsosr 7572 |
Copyright terms: Public domain | W3C validator |