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Theorem prsrriota 7603
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 7598 . . 3 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
2 reurex 2644 . . 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 521 . . . . 5 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
5 simprl 520 . . . . . 6 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → 𝑦P)
6 srpospr 7598 . . . . . . 7 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
76adantr 274 . . . . . 6 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
8 oveq1 5781 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 +P 1P) = (𝑦 +P 1P))
98opeq1d 3711 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨(𝑥 +P 1P), 1P⟩ = ⟨(𝑦 +P 1P), 1P⟩)
109eceq1d 6465 . . . . . . . 8 (𝑥 = 𝑦 → [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨(𝑦 +P 1P), 1P⟩] ~R )
1110eqeq1d 2148 . . . . . . 7 (𝑥 = 𝑦 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴 ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴))
1211riota2 5752 . . . . . 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 408 . . . . 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 146 . . . 4 (((𝐴R ∧ 0R <R 𝐴) ∧ (𝑦P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → (𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦)
15 oveq1 5781 . . . . . 6 ((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦 → ((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P) = (𝑦 +P 1P))
1615opeq1d 3711 . . . . 5 ((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) = 𝑦 → ⟨((𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩ = ⟨(𝑦 +P 1P), 1P⟩)
1716eceq1d 6465 . . . 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 2172 . 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 2554 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 103  wb 104   = wceq 1331  wcel 1480  wrex 2417  ∃!wreu 2418  cop 3530   class class class wbr 3929  crio 5729  (class class class)co 5774  [cec 6427  Pcnp 7106  1Pc1p 7107   +P cpp 7108   ~R cer 7111  Rcnr 7112  0Rc0r 7113   <R cltr 7118
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-i1p 7282  df-iplp 7283  df-iltp 7285  df-enr 7541  df-nr 7542  df-ltr 7545  df-0r 7546
This theorem is referenced by:  caucvgsrlemfv  7606  caucvgsrlemgt1  7610
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