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Theorem pitonnlem1p1 7865
Description: Lemma for pitonn 7867. 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 7573 . . . . . 6  |-  1P  e.  P.
2 addclpr 7556 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
31, 1, 2mp2an 426 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
4 addcomprg 7597 . . . . 5  |-  ( ( A  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( A  +P.  ( 1P  +P.  1P ) )  =  ( ( 1P 
+P.  1P )  +P.  A
) )
53, 4mpan2 425 . . . 4  |-  ( A  e.  P.  ->  ( A  +P.  ( 1P  +P.  1P ) )  =  ( ( 1P  +P.  1P )  +P.  A ) )
65oveq1d 5907 . . 3  |-  ( A  e.  P.  ->  (
( A  +P.  ( 1P  +P.  1P ) )  +P.  1P )  =  ( ( ( 1P 
+P.  1P )  +P.  A
)  +P.  1P )
)
7 addassprg 7598 . . . 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 1339 . . 3  |-  ( A  e.  P.  ->  (
( ( 1P  +P.  1P )  +P.  A )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  1P ) ) )
96, 8eqtrd 2222 . 2  |-  ( A  e.  P.  ->  (
( A  +P.  ( 1P  +P.  1P ) )  +P.  1P )  =  ( ( 1P  +P.  1P )  +P.  ( A  +P.  1P ) ) )
10 addclpr 7556 . . . 4  |-  ( ( A  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( A  +P.  ( 1P  +P.  1P ) )  e.  P. )
113, 10mpan2 425 . . 3  |-  ( A  e.  P.  ->  ( A  +P.  ( 1P  +P.  1P ) )  e.  P. )
123a1i 9 . . 3  |-  ( A  e.  P.  ->  ( 1P  +P.  1P )  e. 
P. )
13 addclpr 7556 . . . 4  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
141, 13mpan2 425 . . 3  |-  ( A  e.  P.  ->  ( A  +P.  1P )  e. 
P. )
151a1i 9 . . 3  |-  ( A  e.  P.  ->  1P  e.  P. )
16 enreceq 7755 . . 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 1250 . 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 167 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 105    = wceq 1364    e. wcel 2160   <.cop 3610  (class class class)co 5892   [cec 6552   P.cnp 7310   1Pc1p 7311    +P. cpp 7312    ~R cer 7315
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-2o 6437  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7323  df-pli 7324  df-mi 7325  df-lti 7326  df-plpq 7363  df-mpq 7364  df-enq 7366  df-nqqs 7367  df-plqqs 7368  df-mqqs 7369  df-1nqqs 7370  df-rq 7371  df-ltnqqs 7372  df-enq0 7443  df-nq0 7444  df-0nq0 7445  df-plq0 7446  df-mq0 7447  df-inp 7485  df-i1p 7486  df-iplp 7487  df-enr 7745
This theorem is referenced by:  pitonnlem2  7866
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