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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 7511 | . . . . 5 ⊢ ~R Er (P × P) | |
2 | erdm 6407 | . . . . 5 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ dom ~R = (P × P) |
4 | ltrelsr 7514 | . . . . . . 7 ⊢ <R ⊆ (R × R) | |
5 | 4 | brel 4561 | . . . . . 6 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → (0R ∈ R ∧ [〈𝐴, 𝐵〉] ~R ∈ R)) |
6 | 5 | simprd 113 | . . . . 5 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → [〈𝐴, 𝐵〉] ~R ∈ R) |
7 | df-nr 7503 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | eleqtrdi 2210 | . . . 4 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → [〈𝐴, 𝐵〉] ~R ∈ ((P × P) / ~R )) |
9 | ecelqsdm 6467 | . . . 4 ⊢ ((dom ~R = (P × P) ∧ [〈𝐴, 𝐵〉] ~R ∈ ((P × P) / ~R )) → 〈𝐴, 𝐵〉 ∈ (P × P)) | |
10 | 3, 8, 9 | sylancr 410 | . . 3 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → 〈𝐴, 𝐵〉 ∈ (P × P)) |
11 | opelxp 4539 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (P × P) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ P)) | |
12 | 10, 11 | sylib 121 | . 2 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
13 | ltrelpr 7281 | . . . 4 ⊢ <P ⊆ (P × P) | |
14 | 13 | brel 4561 | . . 3 ⊢ (𝐵<P 𝐴 → (𝐵 ∈ P ∧ 𝐴 ∈ P)) |
15 | 14 | ancomd 265 | . 2 ⊢ (𝐵<P 𝐴 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
16 | df-0r 7507 | . . . . 5 ⊢ 0R = [〈1P, 1P〉] ~R | |
17 | 16 | breq1i 3906 | . . . 4 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ [〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ) |
18 | 1pr 7330 | . . . . 5 ⊢ 1P ∈ P | |
19 | ltsrprg 7523 | . . . . 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 432 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ([〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
21 | 17, 20 | syl5bb 191 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (0R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
22 | ltaprg 7395 | . . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ 1P ∈ P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) | |
23 | 18, 22 | mp3an3 1289 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
24 | 23 | ancoms 266 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) |
25 | 21, 24 | bitr4d 190 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴)) |
26 | 12, 15, 25 | pm5.21nii 678 | 1 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 〈cop 3500 class class class wbr 3899 × cxp 4507 dom cdm 4509 (class class class)co 5742 Er wer 6394 [cec 6395 / cqs 6396 Pcnp 7067 1Pc1p 7068 +P cpp 7069 <P cltp 7071 ~R cer 7072 Rcnr 7073 0Rc0r 7074 <R cltr 7079 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-i1p 7243 df-iplp 7244 df-iltp 7246 df-enr 7502 df-nr 7503 df-ltr 7506 df-0r 7507 |
This theorem is referenced by: recexgt0sr 7549 mulgt0sr 7554 srpospr 7559 prsrpos 7561 |
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