Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > prsrriota | GIF version |
Description: Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Ref | Expression |
---|---|
prsrriota | ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [〈((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P), 1P〉] ~R = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srpospr 6867 | . . 3 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴) | |
2 | reurex 2523 | . . 3 ⊢ (∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴 → ∃𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴) | |
3 | 1, 2 | syl 14 | . 2 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴) |
4 | simprr 484 | . . . . 5 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴) | |
5 | simprl 483 | . . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → 𝑦 ∈ P) | |
6 | srpospr 6867 | . . . . . . 7 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) | |
7 | 6 | adantr 261 | . . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → ∃!𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) |
8 | oveq1 5519 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 +P 1P) = (𝑦 +P 1P)) | |
9 | 8 | opeq1d 3555 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 〈(𝑥 +P 1P), 1P〉 = 〈(𝑦 +P 1P), 1P〉) |
10 | 9 | eceq1d 6142 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → [〈(𝑥 +P 1P), 1P〉] ~R = [〈(𝑦 +P 1P), 1P〉] ~R ) |
11 | 10 | eqeq1d 2048 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ([〈(𝑥 +P 1P), 1P〉] ~R = 𝐴 ↔ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) |
12 | 11 | riota2 5490 | . . . . . 6 ⊢ ((𝑦 ∈ P ∧ ∃!𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) → ([〈(𝑦 +P 1P), 1P〉] ~R = 𝐴 ↔ (℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) = 𝑦)) |
13 | 5, 7, 12 | syl2anc 391 | . . . . 5 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → ([〈(𝑦 +P 1P), 1P〉] ~R = 𝐴 ↔ (℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) = 𝑦)) |
14 | 4, 13 | mpbid 135 | . . . 4 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → (℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) = 𝑦) |
15 | oveq1 5519 | . . . . . 6 ⊢ ((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) = 𝑦 → ((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P) = (𝑦 +P 1P)) | |
16 | 15 | opeq1d 3555 | . . . . 5 ⊢ ((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) = 𝑦 → 〈((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P), 1P〉 = 〈(𝑦 +P 1P), 1P〉) |
17 | 16 | eceq1d 6142 | . . . 4 ⊢ ((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) = 𝑦 → [〈((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P), 1P〉] ~R = [〈(𝑦 +P 1P), 1P〉] ~R ) |
18 | 14, 17 | syl 14 | . . 3 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → [〈((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P), 1P〉] ~R = [〈(𝑦 +P 1P), 1P〉] ~R ) |
19 | 18, 4 | eqtrd 2072 | . 2 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → [〈((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P), 1P〉] ~R = 𝐴) |
20 | 3, 19 | rexlimddv 2437 | 1 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [〈((℩𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) +P 1P), 1P〉] ~R = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ∃!wreu 2308 〈cop 3378 class class class wbr 3764 ℩crio 5467 (class class class)co 5512 [cec 6104 Pcnp 6389 1Pc1p 6390 +P cpp 6391 ~R cer 6394 Rcnr 6395 0Rc0r 6396 <R cltr 6401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 |
This theorem is referenced by: caucvgsrlemfv 6875 caucvgsrlemgt1 6879 |
Copyright terms: Public domain | W3C validator |