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Theorem prsrlt 6869
Description: Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
Assertion
Ref Expression
prsrlt ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))

Proof of Theorem prsrlt
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pr 6650 . . . . 5 1PP
21a1i 9 . . . 4 ((𝐴P𝐵P) → 1PP)
3 simpr 103 . . . 4 ((𝐴P𝐵P) → 𝐵P)
4 addassprg 6675 . . . 4 ((1PP𝐵P ∧ 1PP) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P)))
52, 3, 2, 4syl3anc 1135 . . 3 ((𝐴P𝐵P) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P)))
65breq2d 3776 . 2 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P) ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P))))
7 simpl 102 . . . 4 ((𝐴P𝐵P) → 𝐴P)
8 ltaprg 6715 . . . 4 ((𝐴P𝐵P ∧ 1PP) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
97, 3, 2, 8syl3anc 1135 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
10 addcomprg 6674 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) = (1P +P 𝐴))
117, 2, 10syl2anc 391 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 1P) = (1P +P 𝐴))
1211breq1d 3774 . . 3 ((𝐴P𝐵P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
13 ltaprg 6715 . . . . 5 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
1413adantl 262 . . . 4 (((𝐴P𝐵P) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
15 addclpr 6633 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
167, 2, 15syl2anc 391 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 1P) ∈ P)
17 addclpr 6633 . . . . 5 ((1PP𝐵P) → (1P +P 𝐵) ∈ P)
182, 3, 17syl2anc 391 . . . 4 ((𝐴P𝐵P) → (1P +P 𝐵) ∈ P)
19 addcomprg 6674 . . . . 5 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2019adantl 262 . . . 4 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2114, 16, 18, 2, 20caovord2d 5670 . . 3 ((𝐴P𝐵P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P)))
229, 12, 213bitr2d 205 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P)))
23 addclpr 6633 . . . 4 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
243, 2, 23syl2anc 391 . . 3 ((𝐴P𝐵P) → (𝐵 +P 1P) ∈ P)
25 ltsrprg 6830 . . 3 ((((𝐴 +P 1P) ∈ P ∧ 1PP) ∧ ((𝐵 +P 1P) ∈ P ∧ 1PP)) → ([⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P))))
2616, 2, 24, 2, 25syl22anc 1136 . 2 ((𝐴P𝐵P) → ([⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P))))
276, 22, 263bitr4d 209 1 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  cop 3378   class class class wbr 3764  (class class class)co 5512  [cec 6104  Pcnp 6387  1Pc1p 6388   +P cpp 6389  <P cltp 6391   ~R cer 6392   <R cltr 6399
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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
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 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813
This theorem is referenced by:  caucvgsrlemcau  6875  caucvgsrlembound  6876  caucvgsrlemgt1  6877  ltrennb  6928
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