Theorem prsrriota 7596

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 7591	. . 3 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
2		reurex 2644	. . 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 521	. . . . 5 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
5		simprl 520	. . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → 𝑦 ∈ P)
6		srpospr 7591	. . . . . . 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 5781	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 +P 1P) = (𝑦 +P 1P))
9	8	opeq1d 3711	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨(𝑥 +P 1P), 1P⟩ = ⟨(𝑦 +P 1P), 1P⟩)
10	9	eceq1d 6465	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨(𝑦 +P 1P), 1P⟩] ~R )
11	10	eqeq1d 2148	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴 ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴))
12	11	riota2 5752	. . . . . 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 408	. . . . 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 5781	. . . . . 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 3711	. . . . 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 6465	. . . 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 2172	. 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 2554	1 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [⟨((℩𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)

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 7099  1_Pc1p 7100   +_P cpp 7101   ~_R cer 7104  Rcnr 7105  0_Rc0r 7106   <_R cltr 7111

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 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-iltp 7278  df-enr 7534  df-nr 7535  df-ltr 7538  df-0r 7539

This theorem is referenced by:  caucvgsrlemfv  7599  caucvgsrlemgt1  7603