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Theorem srpospr 7773
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 7717 . . 3 R = ((P × P) / ~R )
2 breq2 4004 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (0R <R [⟨𝑎, 𝑏⟩] ~R ↔ 0R <R 𝐴))
3 eqeq2 2187 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → ([⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
43reubidv 2660 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝐴 → (∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
52, 4imbi12d 234 . . 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 7738 . . . . . . . 8 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
76biimpi 120 . . . . . . 7 (0R <R [⟨𝑎, 𝑏⟩] ~R𝑏<P 𝑎)
87adantl 277 . . . . . 6 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → 𝑏<P 𝑎)
9 lteupri 7607 . . . . . 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 110 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑥P)
12 1pr 7544 . . . . . . . . . 10 1PP
1312a1i 9 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 1PP)
14 addclpr 7527 . . . . . . . . 9 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) ∈ P)
1511, 13, 14syl2anc 411 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) ∈ P)
16 simplll 533 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑎P)
17 simpllr 534 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → 𝑏P)
18 enreceq 7726 . . . . . . . 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 1239 . . . . . . 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 7568 . . . . . . . . . . . 12 ((𝑥P ∧ 1PP) → (𝑥 +P 1P) = (1P +P 𝑥))
2111, 13, 20syl2anc 411 . . . . . . . . . . 11 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 1P) = (1P +P 𝑥))
2221oveq1d 5884 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = ((1P +P 𝑥) +P 𝑏))
23 addassprg 7569 . . . . . . . . . . 11 ((1PP𝑥P𝑏P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2413, 11, 17, 23syl3anc 1238 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P 𝑥) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2522, 24eqtrd 2210 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 1P) +P 𝑏) = (1P +P (𝑥 +P 𝑏)))
2625eqeq1d 2186 . . . . . . . 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 7527 . . . . . . . . . . 11 ((𝑥P𝑏P) → (𝑥 +P 𝑏) ∈ P)
2811, 17, 27syl2anc 411 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) ∈ P)
29 addcanprg 7606 . . . . . . . . . 10 ((1PP ∧ (𝑥 +P 𝑏) ∈ P𝑎P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
3013, 28, 16, 29syl3anc 1238 . . . . . . . . 9 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) → (𝑥 +P 𝑏) = 𝑎))
31 oveq2 5877 . . . . . . . . 9 ((𝑥 +P 𝑏) = 𝑎 → (1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎))
3230, 31impbid1 142 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((1P +P (𝑥 +P 𝑏)) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
3326, 32bitrd 188 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (((𝑥 +P 1P) +P 𝑏) = (1P +P 𝑎) ↔ (𝑥 +P 𝑏) = 𝑎))
34 addcomprg 7568 . . . . . . . . 9 ((𝑥P𝑏P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3511, 17, 34syl2anc 411 . . . . . . . 8 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → (𝑥 +P 𝑏) = (𝑏 +P 𝑥))
3635eqeq1d 2186 . . . . . . 7 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑥 +P 𝑏) = 𝑎 ↔ (𝑏 +P 𝑥) = 𝑎))
3719, 33, 363bitrrd 215 . . . . . 6 ((((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) ∧ 𝑥P) → ((𝑏 +P 𝑥) = 𝑎 ↔ [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3837reubidva 2659 . . . . 5 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → (∃!𝑥P (𝑏 +P 𝑥) = 𝑎 ↔ ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
3910, 38mpbid 147 . . . 4 (((𝑎P𝑏P) ∧ 0R <R [⟨𝑎, 𝑏⟩] ~R ) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R )
4039ex 115 . . 3 ((𝑎P𝑏P) → (0R <R [⟨𝑎, 𝑏⟩] ~R → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = [⟨𝑎, 𝑏⟩] ~R ))
411, 5, 40ecoptocl 6616 . 2 (𝐴R → (0R <R 𝐴 → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴))
4241imp 124 1 ((𝐴R ∧ 0R <R 𝐴) → ∃!𝑥P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  ∃!wreu 2457  cop 3594   class class class wbr 4000  (class class class)co 5869  [cec 6527  Pcnp 7281  1Pc1p 7282   +P cpp 7283  <P cltp 7285   ~R cer 7286  Rcnr 7287  0Rc0r 7288   <R cltr 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-iltp 7460  df-enr 7716  df-nr 7717  df-ltr 7720  df-0r 7721
This theorem is referenced by:  prsrriota  7778  caucvgsrlemcl  7779  caucvgsrlemgt1  7785
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