Theorem prsrriota 8155

Description: Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Assertion

Ref	Expression
prsrriota	⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [⟨((℩𝑥 ∈ P [⟨(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.

Step	Hyp	Ref	Expression
1		srpospr 8150	. . 3 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
2		reurex 2771	. . 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 537	. . . . 5 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
5		simprl 535	. . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → 𝑦 ∈ P)
6		srpospr 8150	. . . . . . 7 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
7	6	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
8		oveq1 6092	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 +P 1P) = (𝑦 +P 1P))
9	8	opeq1d 3910	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨(𝑥 +P 1P), 1P⟩ = ⟨(𝑦 +P 1P), 1P⟩)
10	9	eceq1d 6843	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨(𝑦 +P 1P), 1P⟩] ~R )
11	10	eqeq1d 2247	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴 ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴))
12	11	riota2 6062	. . . . . 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 415	. . . . 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 147	. . . 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 6092	. . . . . 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 3910	. . . . 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 6843	. . . 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 2271	. 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 2673	1 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [⟨((℩𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ∃!wreu 2530  ⟨cop 3712   class class class wbr 4130  ℩crio 6037  (class class class)co 6085  [cec 6805  Pcnp 7658  1_Pc1p 7659   +_P cpp 7660   ~_R cer 7663  Rcnr 7664  0_Rc0r 7665   <_R cltr 7670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-i1p 7834  df-iplp 7835  df-iltp 7837  df-enr 8093  df-nr 8094  df-ltr 8097  df-0r 8098

This theorem is used by:  caucvgsrlemfv  8158  caucvgsrlemgt1  8162