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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 997 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → 𝐴 ∈ P) | |
2 | 1pr 7531 | . . . 4 ⊢ 1P ∈ P | |
3 | enrex 7714 | . . . . 5 ⊢ ~R ∈ V | |
4 | df-nr 7704 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
5 | 3, 4 | ecopqsi 6583 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → [〈𝐴, 1P〉] ~R ∈ R) |
6 | 1, 2, 5 | sylancl 413 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → [〈𝐴, 1P〉] ~R ∈ R) |
7 | simp2 998 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → 𝐵 ∈ P) | |
8 | 3, 4 | ecopqsi 6583 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → [〈𝐵, 1P〉] ~R ∈ R) |
9 | 7, 2, 8 | sylancl 413 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → [〈𝐵, 1P〉] ~R ∈ R) |
10 | simp3 999 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → 𝐶 ∈ R) | |
11 | ltasrg 7747 | . . 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 1238 | . 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 7555 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) = (1P +P 𝐴)) | |
14 | 1, 2, 13 | sylancl 413 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → (𝐴 +P 1P) = (1P +P 𝐴)) |
15 | 14 | breq1d 4010 | . . 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 7724 | . . . 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 1239 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐴 +P 1P)<P (1P +P 𝐵))) |
19 | ltaprg 7596 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 1P ∈ P) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))) | |
20 | 1, 7, 16, 19 | syl3anc 1238 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))) |
21 | 15, 18, 20 | 3bitr4d 220 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ R) → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ 𝐴<P 𝐵)) |
22 | 12, 21 | bitr3d 190 | 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 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 〈cop 3594 class class class wbr 4000 (class class class)co 5868 [cec 6526 Pcnp 7268 1Pc1p 7269 +P cpp 7270 <P cltp 7272 ~R cer 7273 Rcnr 7274 +R cplr 7278 <R cltr 7280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4285 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-irdg 6364 df-1o 6410 df-2o 6411 df-oadd 6414 df-omul 6415 df-er 6528 df-ec 6530 df-qs 6534 df-ni 7281 df-pli 7282 df-mi 7283 df-lti 7284 df-plpq 7321 df-mpq 7322 df-enq 7324 df-nqqs 7325 df-plqqs 7326 df-mqqs 7327 df-1nqqs 7328 df-rq 7329 df-ltnqqs 7330 df-enq0 7401 df-nq0 7402 df-0nq0 7403 df-plq0 7404 df-mq0 7405 df-inp 7443 df-i1p 7444 df-iplp 7445 df-iltp 7447 df-enr 7703 df-nr 7704 df-plr 7705 df-ltr 7707 |
This theorem is referenced by: suplocsrlemb 7783 suplocsrlempr 7784 suplocsrlem 7785 |
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