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Theorem prsrriota 6872
Description: Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
Assertion
Ref Expression
prsrriota ((𝐴R ∧ 0R <R 𝐴) → [⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem prsrriota
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 srpospr 6867 . . 3 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
2 reurex 2523 . . 3 (∃!𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴 → ∃𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
31, 2syl 14 . 2 ((𝐴R ∧ 0R <R 𝐴) → ∃𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
4 simprr 484 . . . . 5 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
5 simprl 483 . . . . . 6 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → 𝑦P)
6 srpospr 6867 . . . . . . 7 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
76adantr 261 . . . . . 6 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
8 oveq1 5519 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 +P 1P) = (𝑦 +P 1P))
98opeq1d 3555 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨(𝑥 +P 1P), 1P⟩ = ⟨(𝑦 +P 1P), 1P⟩)
109eceq1d 6142 . . . . . . . 8 (𝑥 = 𝑦 → [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨(𝑦 +P 1P), 1P⟩] ~R )
1110eqeq1d 2048 . . . . . . 7 (𝑥 = 𝑦 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴 ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴))
1211riota2 5490 . . . . . 6 ((𝑦P ∧ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) → ([⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴 ↔ (𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦))
135, 7, 12syl2anc 391 . . . . 5 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → ([⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴 ↔ (𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦))
144, 13mpbid 135 . . . 4 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → (𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦)
15 oveq1 5519 . . . . . 6 ((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦 → ((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P) = (𝑦 +P 1P))
1615opeq1d 3555 . . . . 5 ((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦 → ⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩ = ⟨(𝑦 +P 1P), 1P⟩)
1716eceq1d 6142 . . . 4 ((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦 → [⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = [⟨(𝑦 +P 1P), 1P⟩] ~R )
1814, 17syl 14 . . 3 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → [⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = [⟨(𝑦 +P 1P), 1P⟩] ~R )
1918, 4eqtrd 2072 . 2 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → [⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)
203, 19rexlimddv 2437 1 ((𝐴R ∧ 0R <R 𝐴) → [⟨((𝑥P [⟨(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
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