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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  <P cltp 6451   ~R cer 6452  Rcnr 6453  0Rc0r 6454   <R cltr 6459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874
This theorem is referenced by:  prsrriota  6930  caucvgsrlemcl  6931  caucvgsrlemgt1  6937
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