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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
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