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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 6655 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
2 | id 19 | . . . . . . 7 ⊢ ([〈x, y〉] ~R = f → [〈x, y〉] ~R = f) | |
3 | 2, 2 | breq12d 3768 | . . . . . 6 ⊢ ([〈x, y〉] ~R = f → ([〈x, y〉] ~R <R [〈x, y〉] ~R ↔ f <R f)) |
4 | 3 | notbid 591 | . . . . 5 ⊢ ([〈x, y〉] ~R = f → (¬ [〈x, y〉] ~R <R [〈x, y〉] ~R ↔ ¬ f <R f)) |
5 | ltsopr 6570 | . . . . . . . 8 ⊢ <P Or P | |
6 | ltrelpr 6488 | . . . . . . . 8 ⊢ <P ⊆ (P × P) | |
7 | 5, 6 | soirri 4662 | . . . . . . 7 ⊢ ¬ (x +P y)<P (x +P y) |
8 | addcomprg 6554 | . . . . . . . 8 ⊢ ((x ∈ P ∧ y ∈ P) → (x +P y) = (y +P x)) | |
9 | 8 | breq2d 3767 | . . . . . . 7 ⊢ ((x ∈ P ∧ y ∈ P) → ((x +P y)<P (x +P y) ↔ (x +P y)<P (y +P x))) |
10 | 7, 9 | mtbii 598 | . . . . . 6 ⊢ ((x ∈ P ∧ y ∈ P) → ¬ (x +P y)<P (y +P x)) |
11 | ltsrprg 6675 | . . . . . . 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 377 | . . . . . 6 ⊢ ((x ∈ P ∧ y ∈ P) → ([〈x, y〉] ~R <R [〈x, y〉] ~R ↔ (x +P y)<P (y +P x))) |
13 | 10, 12 | mtbird 597 | . . . . 5 ⊢ ((x ∈ P ∧ y ∈ P) → ¬ [〈x, y〉] ~R <R [〈x, y〉] ~R ) |
14 | 1, 4, 13 | ecoptocl 6129 | . . . 4 ⊢ (f ∈ R → ¬ f <R f) |
15 | 14 | adantl 262 | . . 3 ⊢ (( ⊤ ∧ f ∈ R) → ¬ f <R f) |
16 | lttrsr 6690 | . . . 4 ⊢ ((f ∈ R ∧ g ∈ R ∧ ℎ ∈ R) → ((f <R g ∧ g <R ℎ) → f <R ℎ)) | |
17 | 16 | adantl 262 | . . 3 ⊢ (( ⊤ ∧ (f ∈ R ∧ g ∈ R ∧ ℎ ∈ R)) → ((f <R g ∧ g <R ℎ) → f <R ℎ)) |
18 | 15, 17 | ispod 4032 | . 2 ⊢ ( ⊤ → <R Po R) |
19 | 18 | trud 1251 | 1 ⊢ <R Po R |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ⊤ wtru 1243 ∈ wcel 1390 〈cop 3370 class class class wbr 3755 Po wpo 4022 (class class class)co 5455 [cec 6040 Pcnp 6275 +P cpp 6277 <P cltp 6279 ~R cer 6280 Rcnr 6281 <R cltr 6287 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 |
This theorem is referenced by: ltsosr 6692 |
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