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Theorem pitonnlem1p1 7677
Description: Lemma for pitonn 7679. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
Assertion
Ref Expression
pitonnlem1p1  |-  ( A  e.  P.  ->  [ <. ( A  +P.  ( 1P 
+P.  1P ) ) ,  ( 1P  +P.  1P ) >. ]  ~R  =  [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  )

Proof of Theorem pitonnlem1p1
StepHypRef Expression
1 1pr 7385 . . . . . 6  |-  1P  e.  P.
2 addclpr 7368 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
31, 1, 2mp2an 423 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
4 addcomprg 7409 . . . . 5  |-  ( ( A  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( A  +P.  ( 1P  +P.  1P ) )  =  ( ( 1P 
+P.  1P )  +P.  A
) )
53, 4mpan2 422 . . . 4  |-  ( A  e.  P.  ->  ( A  +P.  ( 1P  +P.  1P ) )  =  ( ( 1P  +P.  1P )  +P.  A ) )
65oveq1d 5796 . . 3  |-  ( A  e.  P.  ->  (
( A  +P.  ( 1P  +P.  1P ) )  +P.  1P )  =  ( ( ( 1P 
+P.  1P )  +P.  A
)  +P.  1P )
)
7 addassprg 7410 . . . 4  |-  ( ( ( 1P  +P.  1P )  e.  P.  /\  A  e.  P.  /\  1P  e.  P. )  ->  ( ( ( 1P  +P.  1P )  +P.  A )  +P. 
1P )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  1P ) ) )
83, 1, 7mp3an13 1307 . . 3  |-  ( A  e.  P.  ->  (
( ( 1P  +P.  1P )  +P.  A )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  1P ) ) )
96, 8eqtrd 2173 . 2  |-  ( A  e.  P.  ->  (
( A  +P.  ( 1P  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  1P ) ) )
10 addclpr 7368 . . . 4  |-  ( ( A  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( A  +P.  ( 1P  +P.  1P ) )  e.  P. )
113, 10mpan2 422 . . 3  |-  ( A  e.  P.  ->  ( A  +P.  ( 1P  +P.  1P ) )  e.  P. )
123a1i 9 . . 3  |-  ( A  e.  P.  ->  ( 1P  +P.  1P )  e. 
P. )
13 addclpr 7368 . . . 4  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
141, 13mpan2 422 . . 3  |-  ( A  e.  P.  ->  ( A  +P.  1P )  e. 
P. )
151a1i 9 . . 3  |-  ( A  e.  P.  ->  1P  e.  P. )
16 enreceq 7567 . . 3  |-  ( ( ( ( A  +P.  ( 1P  +P.  1P ) )  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  /\  ( ( A  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. ( A  +P.  ( 1P 
+P.  1P ) ) ,  ( 1P  +P.  1P ) >. ]  ~R  =  [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( A  +P.  ( 1P  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  1P ) ) ) )
1711, 12, 14, 15, 16syl22anc 1218 . 2  |-  ( A  e.  P.  ->  ( [ <. ( A  +P.  ( 1P  +P.  1P ) ) ,  ( 1P 
+P.  1P ) >. ]  ~R  =  [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( A  +P.  ( 1P  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  1P ) ) ) )
189, 17mpbird 166 1  |-  ( A  e.  P.  ->  [ <. ( A  +P.  ( 1P 
+P.  1P ) ) ,  ( 1P  +P.  1P ) >. ]  ~R  =  [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1332    e. wcel 1481   <.cop 3534  (class class class)co 5781   [cec 6434   P.cnp 7122   1Pc1p 7123    +P. cpp 7124    ~R cer 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-i1p 7298  df-iplp 7299  df-enr 7557
This theorem is referenced by:  pitonnlem2  7678
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