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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 6744 | . . . 4 ⊢ 1P ∈ P | |
2 | addclpr 6727 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P) | |
3 | 1, 2 | mpan2 415 | . . 3 ⊢ (𝐴 ∈ P → (𝐴 +P 1P) ∈ P) |
4 | opelxpi 4394 | . . . 4 ⊢ (((𝐴 +P 1P) ∈ P ∧ 1P ∈ P) → 〈(𝐴 +P 1P), 1P〉 ∈ (P × P)) | |
5 | 1, 4 | mpan2 415 | . . 3 ⊢ ((𝐴 +P 1P) ∈ P → 〈(𝐴 +P 1P), 1P〉 ∈ (P × P)) |
6 | enrex 6914 | . . . 4 ⊢ ~R ∈ V | |
7 | 6 | ecelqsi 6183 | . . 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 6904 | . 2 ⊢ R = ((P × P) / ~R ) | |
10 | 8, 9 | syl6eleqr 2172 | 1 ⊢ (𝐴 ∈ P → [〈(𝐴 +P 1P), 1P〉] ~R ∈ R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 〈cop 3401 × cxp 4361 (class class class)co 5532 [cec 6127 / cqs 6128 Pcnp 6481 1Pc1p 6482 +P cpp 6483 ~R cer 6486 Rcnr 6487 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-enr 6903 df-nr 6904 |
This theorem is referenced by: caucvgsrlemgt1 6971 caucvgsrlemoffcau 6974 recidpirq 7026 axcaucvglemcau 7064 |
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