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Theorem pitonnlem1 7621
Description: Lemma for pitonn 7624. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
Assertion
Ref Expression
pitonnlem1  |-  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
Distinct variable group:    u, l

Proof of Theorem pitonnlem1
StepHypRef Expression
1 df-1 7596 . 2  |-  1  =  <. 1R ,  0R >.
2 df-1r 7508 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
3 df-i1p 7243 . . . . . . . 8  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
4 df-1nqqs 7127 . . . . . . . . . . 11  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
54breq2i 3907 . . . . . . . . . 10  |-  ( l 
<Q  1Q  <->  l  <Q  [ <. 1o ,  1o >. ]  ~Q  )
65abbii 2233 . . . . . . . . 9  |-  { l  |  l  <Q  1Q }  =  { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  }
74breq1i 3906 . . . . . . . . . 10  |-  ( 1Q 
<Q  u  <->  [ <. 1o ,  1o >. ]  ~Q  <Q  u
)
87abbii 2233 . . . . . . . . 9  |-  { u  |  1Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u }
96, 8opeq12i 3680 . . . . . . . 8  |-  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
103, 9eqtri 2138 . . . . . . 7  |-  1P  =  <. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
1110oveq1i 5752 . . . . . 6  |-  ( 1P 
+P.  1P )  =  (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
1211opeq1i 3678 . . . . 5  |-  <. ( 1P  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.
13 eceq1 6432 . . . . 5  |-  ( <.
( 1P  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  ->  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
1412, 13ax-mp 5 . . . 4  |-  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R
152, 14eqtri 2138 . . 3  |-  1R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R
1615opeq1i 3678 . 2  |-  <. 1R ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.
171, 16eqtr2i 2139 1  |-  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
Colors of variables: wff set class
Syntax hints:    = wceq 1316   {cab 2103   <.cop 3500   class class class wbr 3899  (class class class)co 5742   1oc1o 6274   [cec 6395    ~Q ceq 7055   1Qc1q 7057    <Q cltq 7061   1Pc1p 7068    +P. cpp 7069    ~R cer 7072   0Rc0r 7074   1Rc1r 7075   1c1 7589
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fv 5101  df-ov 5745  df-ec 6399  df-1nqqs 7127  df-i1p 7243  df-1r 7508  df-1 7596
This theorem is referenced by:  pitonn  7624
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