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Mirrors > Home > ILE Home > Th. List > ltpsrprg | GIF version |
Description: Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
Ref | Expression |
---|---|
ltpsrprg | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ((𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ) ↔ 𝐴<P 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → 𝐴 ∈ P) | |
2 | 1pr 7365 | . . . 4 ⊢ 1P ∈ P | |
3 | enrex 7548 | . . . . 5 ⊢ ~R ∈ V | |
4 | df-nr 7538 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
5 | 3, 4 | ecopqsi 6484 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → [〈𝐴, 1P〉] ~R ∈ R) |
6 | 1, 2, 5 | sylancl 409 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → [〈𝐴, 1P〉] ~R ∈ R) |
7 | simp2 982 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → 𝐵 ∈ P) | |
8 | 3, 4 | ecopqsi 6484 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → [〈𝐵, 1P〉] ~R ∈ R) |
9 | 7, 2, 8 | sylancl 409 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → [〈𝐵, 1P〉] ~R ∈ R) |
10 | simp3 983 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → 𝐶 ∈ R) | |
11 | ltasrg 7581 | . . 3 ⊢ (([〈𝐴, 1P〉] ~R ∈ R ∧ [〈𝐵, 1P〉] ~R ∈ R ∧ 𝐶 ∈ R) → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ))) | |
12 | 6, 9, 10, 11 | syl3anc 1216 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ))) |
13 | addcomprg 7389 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) = (1P +P 𝐴)) | |
14 | 1, 2, 13 | sylancl 409 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → (𝐴 +P 1P) = (1P +P 𝐴)) |
15 | 14 | breq1d 3939 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵))) |
16 | 2 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → 1P ∈ P) |
17 | ltsrprg 7558 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 1P ∈ P) ∧ (𝐵 ∈ P ∧ 1P ∈ P)) → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐴 +P 1P)<P (1P +P 𝐵))) | |
18 | 1, 16, 7, 16, 17 | syl22anc 1217 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐴 +P 1P)<P (1P +P 𝐵))) |
19 | ltaprg 7430 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 1P ∈ P) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))) | |
20 | 1, 7, 16, 19 | syl3anc 1216 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))) |
21 | 15, 18, 20 | 3bitr4d 219 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ 𝐴<P 𝐵)) |
22 | 12, 21 | bitr3d 189 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ((𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ) ↔ 𝐴<P 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 (class class class)co 5774 [cec 6427 Pcnp 7102 1Pc1p 7103 +P cpp 7104 <P cltp 7106 ~R cer 7107 Rcnr 7108 +R cplr 7112 <R cltr 7114 |
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 7115 df-pli 7116 df-mi 7117 df-lti 7118 df-plpq 7155 df-mpq 7156 df-enq 7158 df-nqqs 7159 df-plqqs 7160 df-mqqs 7161 df-1nqqs 7162 df-rq 7163 df-ltnqqs 7164 df-enq0 7235 df-nq0 7236 df-0nq0 7237 df-plq0 7238 df-mq0 7239 df-inp 7277 df-i1p 7278 df-iplp 7279 df-iltp 7281 df-enr 7537 df-nr 7538 df-plr 7539 df-ltr 7541 |
This theorem is referenced by: suplocsrlemb 7617 suplocsrlempr 7618 suplocsrlem 7619 |
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