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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 7615 | . . 3 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑦 ∈ P [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴) | |
2 | reurex 2647 | . . 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 522 | . . . . 5 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴) | |
5 | simprl 521 | . . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → 𝑦 ∈ P) | |
6 | srpospr 7615 | . . . . . . 7 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) | |
7 | 6 | adantr 274 | . . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) → ∃!𝑥 ∈ P [〈(𝑥 +P 1P), 1P〉] ~R = 𝐴) |
8 | oveq1 5789 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 +P 1P) = (𝑦 +P 1P)) | |
9 | 8 | opeq1d 3719 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 〈(𝑥 +P 1P), 1P〉 = 〈(𝑦 +P 1P), 1P〉) |
10 | 9 | eceq1d 6473 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → [〈(𝑥 +P 1P), 1P〉] ~R = [〈(𝑦 +P 1P), 1P〉] ~R ) |
11 | 10 | eqeq1d 2149 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ([〈(𝑥 +P 1P), 1P〉] ~R = 𝐴 ↔ [〈(𝑦 +P 1P), 1P〉] ~R = 𝐴)) |
12 | 11 | riota2 5760 | . . . . . 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 409 | . . . . 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 146 | . . . 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 5789 | . . . . . 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 3719 | . . . . 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 6473 | . . . 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 2173 | . 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 2557 | 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 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 ∃!wreu 2419 〈cop 3535 class class class wbr 3937 ℩crio 5737 (class class class)co 5782 [cec 6435 Pcnp 7123 1Pc1p 7124 +P cpp 7125 ~R cer 7128 Rcnr 7129 0Rc0r 7130 <R cltr 7135 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-iltp 7302 df-enr 7558 df-nr 7559 df-ltr 7562 df-0r 7563 |
This theorem is referenced by: caucvgsrlemfv 7623 caucvgsrlemgt1 7627 |
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