Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > gt0srpr | GIF version |
Description: Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Ref | Expression |
---|---|
gt0srpr | ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrer 6820 | . . . . 5 ⊢ ~R Er (P × P) | |
2 | erdm 6116 | . . . . 5 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
3 | 1, 2 | ax-mp 7 | . . . 4 ⊢ dom ~R = (P × P) |
4 | ltrelsr 6823 | . . . . . . 7 ⊢ <R ⊆ (R × R) | |
5 | 4 | brel 4392 | . . . . . 6 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → (0R ∈ R ∧ [〈𝐴, 𝐵〉] ~R ∈ R)) |
6 | 5 | simprd 107 | . . . . 5 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → [〈𝐴, 𝐵〉] ~R ∈ R) |
7 | df-nr 6812 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | syl6eleq 2130 | . . . 4 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → [〈𝐴, 𝐵〉] ~R ∈ ((P × P) / ~R )) |
9 | ecelqsdm 6176 | . . . 4 ⊢ ((dom ~R = (P × P) ∧ [〈𝐴, 𝐵〉] ~R ∈ ((P × P) / ~R )) → 〈𝐴, 𝐵〉 ∈ (P × P)) | |
10 | 3, 8, 9 | sylancr 393 | . . 3 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → 〈𝐴, 𝐵〉 ∈ (P × P)) |
11 | opelxp 4374 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (P × P) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ P)) | |
12 | 10, 11 | sylib 127 | . 2 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
13 | ltrelpr 6603 | . . . 4 ⊢ <P ⊆ (P × P) | |
14 | 13 | brel 4392 | . . 3 ⊢ (𝐵<P 𝐴 → (𝐵 ∈ P ∧ 𝐴 ∈ P)) |
15 | 14 | ancomd 254 | . 2 ⊢ (𝐵<P 𝐴 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
16 | df-0r 6816 | . . . . 5 ⊢ 0R = [〈1P, 1P〉] ~R | |
17 | 16 | breq1i 3771 | . . . 4 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ [〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ) |
18 | 1pr 6652 | . . . . 5 ⊢ 1P ∈ P | |
19 | ltsrprg 6832 | . . . . 5 ⊢ (((1P ∈ P ∧ 1P ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) → ([〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴))) | |
20 | 18, 18, 19 | mpanl12 412 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ([〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
21 | 17, 20 | syl5bb 181 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (0R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
22 | ltaprg 6717 | . . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ 1P ∈ P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) | |
23 | 18, 22 | mp3an3 1221 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
24 | 23 | ancoms 255 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
25 | 21, 24 | bitr4d 180 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴)) |
26 | 12, 15, 25 | pm5.21nii 620 | 1 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 〈cop 3378 class class class wbr 3764 × cxp 4343 dom cdm 4345 (class class class)co 5512 Er wer 6103 [cec 6104 / cqs 6105 Pcnp 6389 1Pc1p 6390 +P cpp 6391 <P cltp 6393 ~R cer 6394 Rcnr 6395 0Rc0r 6396 <R cltr 6401 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 |
This theorem is referenced by: recexgt0sr 6858 mulgt0sr 6862 srpospr 6867 prsrpos 6869 |
Copyright terms: Public domain | W3C validator |