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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 7659 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
2 | id 19 | . . . . . . 7 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝑓 → [〈𝑥, 𝑦〉] ~R = 𝑓) | |
3 | 2, 2 | breq12d 3989 | . . . . . 6 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝑓 → ([〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ↔ 𝑓 <R 𝑓)) |
4 | 3 | notbid 657 | . . . . 5 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝑓 → (¬ [〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ↔ ¬ 𝑓 <R 𝑓)) |
5 | ltsopr 7528 | . . . . . . . 8 ⊢ <P Or P | |
6 | ltrelpr 7437 | . . . . . . . 8 ⊢ <P ⊆ (P × P) | |
7 | 5, 6 | soirri 4992 | . . . . . . 7 ⊢ ¬ (𝑥 +P 𝑦)<P (𝑥 +P 𝑦) |
8 | addcomprg 7510 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥)) | |
9 | 8 | breq2d 3988 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 𝑦)<P (𝑥 +P 𝑦) ↔ (𝑥 +P 𝑦)<P (𝑦 +P 𝑥))) |
10 | 7, 9 | mtbii 664 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ¬ (𝑥 +P 𝑦)<P (𝑦 +P 𝑥)) |
11 | ltsrprg 7679 | . . . . . . 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 395 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ↔ (𝑥 +P 𝑦)<P (𝑦 +P 𝑥))) |
13 | 10, 12 | mtbird 663 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ¬ [〈𝑥, 𝑦〉] ~R <R [〈𝑥, 𝑦〉] ~R ) |
14 | 1, 4, 13 | ecoptocl 6579 | . . . 4 ⊢ (𝑓 ∈ R → ¬ 𝑓 <R 𝑓) |
15 | 14 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑓 ∈ R) → ¬ 𝑓 <R 𝑓) |
16 | lttrsr 7694 | . . . 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 4276 | . 2 ⊢ (⊤ → <R Po R) |
19 | 18 | mptru 1351 | 1 ⊢ <R Po R |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 = wceq 1342 ⊤wtru 1343 ∈ wcel 2135 〈cop 3573 class class class wbr 3976 Po wpo 4266 (class class class)co 5836 [cec 6490 Pcnp 7223 +P cpp 7225 <P cltp 7227 ~R cer 7228 Rcnr 7229 <R cltr 7235 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-iplp 7400 df-iltp 7402 df-enr 7658 df-nr 7659 df-ltr 7662 |
This theorem is referenced by: ltsosr 7696 |
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