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Mirrors > Home > ILE Home > Th. List > prsrcl | GIF version |
Description: Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Ref | Expression |
---|---|
prsrcl | ⊢ (𝐴 ∈ P → [〈(𝐴 +P 1P), 1P〉] ~R ∈ R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 7310 | . . . 4 ⊢ 1P ∈ P | |
2 | addclpr 7293 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P) | |
3 | 1, 2 | mpan2 419 | . . 3 ⊢ (𝐴 ∈ P → (𝐴 +P 1P) ∈ P) |
4 | opelxpi 4531 | . . . 4 ⊢ (((𝐴 +P 1P) ∈ P ∧ 1P ∈ P) → 〈(𝐴 +P 1P), 1P〉 ∈ (P × P)) | |
5 | 1, 4 | mpan2 419 | . . 3 ⊢ ((𝐴 +P 1P) ∈ P → 〈(𝐴 +P 1P), 1P〉 ∈ (P × P)) |
6 | enrex 7480 | . . . 4 ⊢ ~R ∈ V | |
7 | 6 | ecelqsi 6437 | . . 3 ⊢ (〈(𝐴 +P 1P), 1P〉 ∈ (P × P) → [〈(𝐴 +P 1P), 1P〉] ~R ∈ ((P × P) / ~R )) |
8 | 3, 5, 7 | 3syl 17 | . 2 ⊢ (𝐴 ∈ P → [〈(𝐴 +P 1P), 1P〉] ~R ∈ ((P × P) / ~R )) |
9 | df-nr 7470 | . 2 ⊢ R = ((P × P) / ~R ) | |
10 | 8, 9 | syl6eleqr 2208 | 1 ⊢ (𝐴 ∈ P → [〈(𝐴 +P 1P), 1P〉] ~R ∈ R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1463 〈cop 3496 × cxp 4497 (class class class)co 5728 [cec 6381 / cqs 6382 Pcnp 7047 1Pc1p 7048 +P cpp 7049 ~R cer 7052 Rcnr 7053 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-eprel 4171 df-id 4175 df-po 4178 df-iso 4179 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-recs 6156 df-irdg 6221 df-1o 6267 df-2o 6268 df-oadd 6271 df-omul 6272 df-er 6383 df-ec 6385 df-qs 6389 df-ni 7060 df-pli 7061 df-mi 7062 df-lti 7063 df-plpq 7100 df-mpq 7101 df-enq 7103 df-nqqs 7104 df-plqqs 7105 df-mqqs 7106 df-1nqqs 7107 df-rq 7108 df-ltnqqs 7109 df-enq0 7180 df-nq0 7181 df-0nq0 7182 df-plq0 7183 df-mq0 7184 df-inp 7222 df-i1p 7223 df-iplp 7224 df-enr 7469 df-nr 7470 |
This theorem is referenced by: caucvgsrlemgt1 7537 caucvgsrlemoffcau 7540 recidpirq 7593 axcaucvglemcau 7633 |
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