ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prsrlt GIF version

Theorem prsrlt 7785
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 7552 . . . . 5 1PP
21a1i 9 . . . 4 ((𝐴P𝐵P) → 1PP)
3 simpr 110 . . . 4 ((𝐴P𝐵P) → 𝐵P)
4 addassprg 7577 . . . 4 ((1PP𝐵P ∧ 1PP) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P)))
52, 3, 2, 4syl3anc 1238 . . 3 ((𝐴P𝐵P) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P)))
65breq2d 4015 . 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 109 . . . 4 ((𝐴P𝐵P) → 𝐴P)
8 ltaprg 7617 . . . 4 ((𝐴P𝐵P ∧ 1PP) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
97, 3, 2, 8syl3anc 1238 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
10 addcomprg 7576 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) = (1P +P 𝐴))
117, 2, 10syl2anc 411 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 1P) = (1P +P 𝐴))
1211breq1d 4013 . . 3 ((𝐴P𝐵P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
13 ltaprg 7617 . . . . 5 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
1413adantl 277 . . . 4 (((𝐴P𝐵P) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
15 addclpr 7535 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
167, 2, 15syl2anc 411 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 1P) ∈ P)
17 addclpr 7535 . . . . 5 ((1PP𝐵P) → (1P +P 𝐵) ∈ P)
182, 3, 17syl2anc 411 . . . 4 ((𝐴P𝐵P) → (1P +P 𝐵) ∈ P)
19 addcomprg 7576 . . . . 5 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2019adantl 277 . . . 4 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2114, 16, 18, 2, 20caovord2d 6043 . . 3 ((𝐴P𝐵P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P)))
229, 12, 213bitr2d 216 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P)))
23 addclpr 7535 . . . 4 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
243, 2, 23syl2anc 411 . . 3 ((𝐴P𝐵P) → (𝐵 +P 1P) ∈ P)
25 ltsrprg 7745 . . 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 1239 . 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 220 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 104  wb 105  w3a 978   = wceq 1353  wcel 2148  cop 3595   class class class wbr 4003  (class class class)co 5874  [cec 6532  Pcnp 7289  1Pc1p 7290   +P cpp 7291  <P cltp 7293   ~R cer 7294   <R cltr 7301
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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-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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-iplp 7466  df-iltp 7468  df-enr 7724  df-nr 7725  df-ltr 7728
This theorem is referenced by:  caucvgsrlemcau  7791  caucvgsrlembound  7792  caucvgsrlemgt1  7793  ltrennb  7852
  Copyright terms: Public domain W3C validator