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Theorem prsrlt 7061
Description: Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
Assertion
Ref Expression
prsrlt  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  <R  [
<. ( B  +P.  1P ) ,  1P >. ]  ~R  ) )

Proof of Theorem prsrlt
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pr 6842 . . . . 5  |-  1P  e.  P.
21a1i 9 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  1P  e.  P. )
3 simpr 108 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  B  e.  P. )
4 addassprg 6867 . . . 4  |-  ( ( 1P  e.  P.  /\  B  e.  P.  /\  1P  e.  P. )  ->  (
( 1P  +P.  B
)  +P.  1P )  =  ( 1P  +P.  ( B  +P.  1P ) ) )
52, 3, 2, 4syl3anc 1170 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( 1P  +P.  B )  +P.  1P )  =  ( 1P  +P.  ( B  +P.  1P ) ) )
65breq2d 3818 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( A  +P.  1P )  +P. 
1P )  <P  (
( 1P  +P.  B
)  +P.  1P )  <->  ( ( A  +P.  1P )  +P.  1P )  <P 
( 1P  +P.  ( B  +P.  1P ) ) ) )
7 simpl 107 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  e.  P. )
8 ltaprg 6907 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  1P  e.  P. )  ->  ( A  <P  B  <->  ( 1P  +P.  A )  <P  ( 1P  +P.  B ) ) )
97, 3, 2, 8syl3anc 1170 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  ( 1P  +P.  A )  <P  ( 1P  +P.  B ) ) )
10 addcomprg 6866 . . . . 5  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  =  ( 1P  +P.  A ) )
117, 2, 10syl2anc 403 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  1P )  =  ( 1P  +P.  A ) )
1211breq1d 3816 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  <P  ( 1P  +P.  B )  <->  ( 1P  +P.  A )  <P  ( 1P  +P.  B ) ) )
13 ltaprg 6907 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1413adantl 271 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )
)  ->  ( f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
15 addclpr 6825 . . . . 5  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
167, 2, 15syl2anc 403 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  1P )  e.  P. )
17 addclpr 6825 . . . . 5  |-  ( ( 1P  e.  P.  /\  B  e.  P. )  ->  ( 1P  +P.  B
)  e.  P. )
182, 3, 17syl2anc 403 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 1P  +P.  B
)  e.  P. )
19 addcomprg 6866 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2019adantl 271 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
2114, 16, 18, 2, 20caovord2d 5722 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  1P )  <P  ( 1P  +P.  B )  <->  ( ( A  +P.  1P )  +P. 
1P )  <P  (
( 1P  +P.  B
)  +P.  1P )
) )
229, 12, 213bitr2d 214 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  ( ( A  +P.  1P )  +P. 
1P )  <P  (
( 1P  +P.  B
)  +P.  1P )
) )
23 addclpr 6825 . . . 4  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  +P.  1P )  e.  P. )
243, 2, 23syl2anc 403 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( B  +P.  1P )  e.  P. )
25 ltsrprg 7022 . . 3  |-  ( ( ( ( A  +P.  1P )  e.  P.  /\  1P  e.  P. )  /\  ( ( B  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. ( 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 1171 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( [ <. ( 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 218 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( 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 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3420   class class class wbr 3806  (class class class)co 5564   [cec 6192   P.cnp 6579   1Pc1p 6580    +P. cpp 6581    <P cltp 6583    ~R cer 6584    <R cltr 6591
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 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
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-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641  df-enq0 6712  df-nq0 6713  df-0nq0 6714  df-plq0 6715  df-mq0 6716  df-inp 6754  df-i1p 6755  df-iplp 6756  df-iltp 6758  df-enr 7001  df-nr 7002  df-ltr 7005
This theorem is referenced by:  caucvgsrlemcau  7067  caucvgsrlembound  7068  caucvgsrlemgt1  7069  ltrennb  7120
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