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Theorem srpospr 7229
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 7174 . . 3 R = ((P × P) / ~R )
2 breq2 3815 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3 eqeq2 2092 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
43reubidv 2543 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
52, 4imbi12d 232 . . 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 7195 . . . . . . . 8 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
76biimpi 118 . . . . . . 7 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
87adantl 271 . . . . . 6 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → 𝑏<P 𝑎)
9 lteupri 7077 . . . . . 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 108 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑥P)
12 1pr 7014 . . . . . . . . . 10 1PP
1312a1i 9 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 1PP)
14 addclpr 6997 . . . . . . . . 9 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) ∈ P)
1511, 13, 14syl2anc 403 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) ∈ P)
16 simplll 500 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑎P)
17 simpllr 501 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑏P)
18 enreceq 7183 . . . . . . . 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 1171 . . . . . . 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 7038 . . . . . . . . . . . 12 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) = (1P +P 𝑥))
2111, 13, 20syl2anc 403 . . . . . . . . . . 11 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) = (1P +P 𝑥))
2221oveq1d 5604 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23 addassprg 7039 . . . . . . . . . . 11 ((1PP𝑥P𝑏P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2413, 11, 17, 23syl3anc 1170 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2522, 24eqtrd 2115 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2625eqeq1d 2091 . . . . . . . 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 6997 . . . . . . . . . . 11 ((𝑥P𝑏P) → (𝑥 +P 𝑏) ∈ P)
2811, 17, 27syl2anc 403 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) ∈ P)
29 addcanprg 7076 . . . . . . . . . 10 ((1PP ∧ (𝑥 +P 𝑏) ∈ P𝑎P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
3013, 28, 16, 29syl3anc 1170 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
31 oveq2 5597 . . . . . . . . 9 ((𝑥 +P 𝑏) = 𝑎 → (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎))
3230, 31impbid1 140 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
3326, 32bitrd 186 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
34 addcomprg 7038 . . . . . . . . 9 ((𝑥P𝑏P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3511, 17, 34syl2anc 403 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3635eqeq1d 2091 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 𝑏) = 𝑎 ↔ (𝑏 +P 𝑥) = 𝑎))
3719, 33, 363bitrrd 213 . . . . . 6 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑏 +P 𝑥) = 𝑎 ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3837reubidva 2542 . . . . 5 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → (∃!𝑥P (𝑏 +P 𝑥) = 𝑎 ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3910, 38mpbid 145 . . . 4 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R )
4039ex 113 . . 3 ((𝑎P𝑏P) → (0R <R [⟨𝑎, 𝑏⟩] ~R → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
411, 5, 40ecoptocl 6307 . 2 (𝐴R → (0R <R 𝐴 → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4241imp 122 1 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  ∃!wreu 2355  cop 3425   class class class wbr 3811  (class class class)co 5589  [cec 6218  Pcnp 6751  1Pc1p 6752   +P cpp 6753  <P cltp 6755   ~R cer 6756  Rcnr 6757  0Rc0r 6758   <R cltr 6763
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4079  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-1o 6111  df-2o 6112  df-oadd 6115  df-omul 6116  df-er 6220  df-ec 6222  df-qs 6226  df-ni 6764  df-pli 6765  df-mi 6766  df-lti 6767  df-plpq 6804  df-mpq 6805  df-enq 6807  df-nqqs 6808  df-plqqs 6809  df-mqqs 6810  df-1nqqs 6811  df-rq 6812  df-ltnqqs 6813  df-enq0 6884  df-nq0 6885  df-0nq0 6886  df-plq0 6887  df-mq0 6888  df-inp 6926  df-i1p 6927  df-iplp 6928  df-iltp 6930  df-enr 7173  df-nr 7174  df-ltr 7177  df-0r 7178
This theorem is referenced by:  prsrriota  7234  caucvgsrlemcl  7235  caucvgsrlemgt1  7241
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