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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 7543 | . . . . 5 ⊢ ~R Er (P × P) | |
2 | erdm 6439 | . . . . 5 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ dom ~R = (P × P) |
4 | ltrelsr 7546 | . . . . . . 7 ⊢ <R ⊆ (R × R) | |
5 | 4 | brel 4591 | . . . . . 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 7535 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | eleqtrdi 2232 | . . . 4 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R → [〈𝐴, 𝐵〉] ~R ∈ ((P × P) / ~R )) |
9 | ecelqsdm 6499 | . . . 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 4569 | . . 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 7313 | . . . 4 ⊢ <P ⊆ (P × P) | |
14 | 13 | brel 4591 | . . 3 ⊢ (𝐵<P 𝐴 → (𝐵 ∈ P ∧ 𝐴 ∈ P)) |
15 | 14 | ancomd 265 | . 2 ⊢ (𝐵<P 𝐴 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
16 | df-0r 7539 | . . . . 5 ⊢ 0R = [〈1P, 1P〉] ~R | |
17 | 16 | breq1i 3936 | . . . 4 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ [〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ) |
18 | 1pr 7362 | . . . . 5 ⊢ 1P ∈ P | |
19 | ltsrprg 7555 | . . . . 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 7427 | . . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ 1P ∈ P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) | |
23 | 18, 22 | mp3an3 1304 | . . . 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 693 | 1 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 × cxp 4537 dom cdm 4539 (class class class)co 5774 Er wer 6426 [cec 6427 / cqs 6428 Pcnp 7099 1Pc1p 7100 +P cpp 7101 <P cltp 7103 ~R cer 7104 Rcnr 7105 0Rc0r 7106 <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-i1p 7275 df-iplp 7276 df-iltp 7278 df-enr 7534 df-nr 7535 df-ltr 7538 df-0r 7539 |
This theorem is referenced by: recexgt0sr 7581 mulgt0sr 7586 srpospr 7591 prsrpos 7593 |
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