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Theorem srpospr 6824
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 6769 . . 3 R = ((P × P) / ~R )
2 breq2 3765 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3 eqeq2 2049 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
43reubidv 2490 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
52, 4imbi12d 223 . . 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 6790 . . . . . . . 8 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
76biimpi 113 . . . . . . 7 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
87adantl 262 . . . . . 6 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → 𝑏<P 𝑎)
9 lteupri 6672 . . . . . 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 103 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑥P)
12 1pr 6609 . . . . . . . . . 10 1PP
1312a1i 9 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 1PP)
14 addclpr 6592 . . . . . . . . 9 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) ∈ P)
1511, 13, 14syl2anc 391 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) ∈ P)
16 simplll 485 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑎P)
17 simpllr 486 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑏P)
18 enreceq 6778 . . . . . . . 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 1136 . . . . . . 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 6633 . . . . . . . . . . . 12 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) = (1P +P 𝑥))
2111, 13, 20syl2anc 391 . . . . . . . . . . 11 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) = (1P +P 𝑥))
2221oveq1d 5490 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23 addassprg 6634 . . . . . . . . . . 11 ((1PP𝑥P𝑏P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2413, 11, 17, 23syl3anc 1135 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2522, 24eqtrd 2072 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2625eqeq1d 2048 . . . . . . . 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 6592 . . . . . . . . . . 11 ((𝑥P𝑏P) → (𝑥 +P 𝑏) ∈ P)
2811, 17, 27syl2anc 391 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) ∈ P)
29 addcanprg 6671 . . . . . . . . . 10 ((1PP ∧ (𝑥 +P 𝑏) ∈ P𝑎P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
3013, 28, 16, 29syl3anc 1135 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
31 oveq2 5483 . . . . . . . . 9 ((𝑥 +P 𝑏) = 𝑎 → (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎))
3230, 31impbid1 130 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
3326, 32bitrd 177 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
34 addcomprg 6633 . . . . . . . . 9 ((𝑥P𝑏P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3511, 17, 34syl2anc 391 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3635eqeq1d 2048 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 𝑏) = 𝑎 ↔ (𝑏 +P 𝑥) = 𝑎))
3719, 33, 363bitrrd 204 . . . . . 6 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑏 +P 𝑥) = 𝑎 ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3837reubidva 2489 . . . . 5 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → (∃!𝑥P (𝑏 +P 𝑥) = 𝑎 ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3910, 38mpbid 135 . . . 4 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R )
4039ex 108 . . 3 ((𝑎P𝑏P) → (0R <R [⟨𝑎, 𝑏⟩] ~R → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
411, 5, 40ecoptocl 6156 . 2 (𝐴R → (0R <R 𝐴 → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4241imp 115 1 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  ∃!wreu 2305  cop 3375   class class class wbr 3761  (class class class)co 5475  [cec 6067  Pcnp 6346  1Pc1p 6347   +P cpp 6348  <P cltp 6350   ~R cer 6351  Rcnr 6352  0Rc0r 6353   <R cltr 6358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rmo 2311  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-i1p 6522  df-iplp 6523  df-iltp 6525  df-enr 6768  df-nr 6769  df-ltr 6772  df-0r 6773
This theorem is referenced by:  prsrriota  6829  caucvgsrlemcl  6830  caucvgsrlemgt1  6836
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