Theorem prsrriota 6964

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 6959	. . 3 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
2		reurex 2567	. . 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 498	. . . . 5 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)
5		simprl 497	. . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → 𝑦 ∈ P)
6		srpospr 6959	. . . . . . 7 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
7	6	adantr 270	. . . . . 6 ⊢ (((𝐴 ∈ R ∧ 0R <R 𝐴) ∧ (𝑦 ∈ P ∧ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴)) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
8		oveq1 5539	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 +P 1P) = (𝑦 +P 1P))
9	8	opeq1d 3576	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨(𝑥 +P 1P), 1P⟩ = ⟨(𝑦 +P 1P), 1P⟩)
10	9	eceq1d 6165	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨(𝑦 +P 1P), 1P⟩] ~R )
11	10	eqeq1d 2089	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴 ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = 𝐴))
12	11	riota2 5510	. . . . . 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 403	. . . . 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 145	. . . 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 5539	. . . . . 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 3576	. . . . 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 6165	. . . 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 2113	. 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 2481	1 ⊢ ((𝐴 ∈ R ∧ 0R <R 𝐴) → [⟨((℩𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∃wrex 2349  ∃!wreu 2350  ⟨cop 3401   class class class wbr 3785  ℩crio 5487  (class class class)co 5532  [cec 6127  Pcnp 6481  1_Pc1p 6482   +_P cpp 6483   ~_R cer 6486  Rcnr 6487  0_Rc0r 6488   <_R cltr 6493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-iltp 6660  df-enr 6903  df-nr 6904  df-ltr 6907  df-0r 6908

This theorem is referenced by:  caucvgsrlemfv  6967  caucvgsrlemgt1  6971