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Theorem srpospr 6925
 Description: Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
Assertion
Ref Expression
srpospr ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem srpospr
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 6870 . . 3 R = ((P × P) / ~R )
2 breq2 3796 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3 eqeq2 2065 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
43reubidv 2510 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
52, 4imbi12d 227 . . 3 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ((0R <R [⟨𝑎, 𝑏⟩] ~R → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ) ↔ (0R <R 𝐴 → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)))
6 gt0srpr 6891 . . . . . . . 8 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
76biimpi 117 . . . . . . 7 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
87adantl 266 . . . . . 6 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → 𝑏<P 𝑎)
9 lteupri 6773 . . . . . 6 (𝑏<P 𝑎 → ∃!𝑥P (𝑏 +P 𝑥) = 𝑎)
108, 9syl 14 . . . . 5 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥P (𝑏 +P 𝑥) = 𝑎)
11 simpr 107 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑥P)
12 1pr 6710 . . . . . . . . . 10 1PP
1312a1i 9 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 1PP)
14 addclpr 6693 . . . . . . . . 9 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) ∈ P)
1511, 13, 14syl2anc 397 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) ∈ P)
16 simplll 493 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑎P)
17 simpllr 494 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑏P)
18 enreceq 6879 . . . . . . . 8 ((((𝑥 +P 1P) ∈ P ∧ 1PP) ∧ (𝑎P𝑏P)) → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎)))
1915, 13, 16, 17, 18syl22anc 1147 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎)))
20 addcomprg 6734 . . . . . . . . . . . 12 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) = (1P +P 𝑥))
2111, 13, 20syl2anc 397 . . . . . . . . . . 11 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) = (1P +P 𝑥))
2221oveq1d 5555 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23 addassprg 6735 . . . . . . . . . . 11 ((1PP𝑥P𝑏P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2413, 11, 17, 23syl3anc 1146 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2522, 24eqtrd 2088 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2625eqeq1d 2064 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎) ↔ (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎)))
27 addclpr 6693 . . . . . . . . . . 11 ((𝑥P𝑏P) → (𝑥 +P 𝑏) ∈ P)
2811, 17, 27syl2anc 397 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) ∈ P)
29 addcanprg 6772 . . . . . . . . . 10 ((1PP ∧ (𝑥 +P 𝑏) ∈ P𝑎P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
3013, 28, 16, 29syl3anc 1146 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
31 oveq2 5548 . . . . . . . . 9 ((𝑥 +P 𝑏) = 𝑎 → (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎))
3230, 31impbid1 134 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
3326, 32bitrd 181 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
34 addcomprg 6734 . . . . . . . . 9 ((𝑥P𝑏P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3511, 17, 34syl2anc 397 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3635eqeq1d 2064 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 𝑏) = 𝑎 ↔ (𝑏 +P 𝑥) = 𝑎))
3719, 33, 363bitrrd 208 . . . . . 6 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑏 +P 𝑥) = 𝑎 ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3837reubidva 2509 . . . . 5 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → (∃!𝑥P (𝑏 +P 𝑥) = 𝑎 ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3910, 38mpbid 139 . . . 4 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R )
4039ex 112 . . 3 ((𝑎P𝑏P) → (0R <R [⟨𝑎, 𝑏⟩] ~R → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
411, 5, 40ecoptocl 6224 . 2 (𝐴R → (0R <R 𝐴 → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4241imp 119 1 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  ∃!wreu 2325  ⟨cop 3406   class class class wbr 3792  (class class class)co 5540  [cec 6135  Pcnp 6447  1Pc1p 6448   +P cpp 6449
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