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Theorem pitonnlem1 7445
Description: Lemma for pitonn 7448. 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 7421 . 2  |-  1  =  <. 1R ,  0R >.
2 df-1r 7341 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
3 df-i1p 7089 . . . . . . . 8  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
4 df-1nqqs 6973 . . . . . . . . . . 11  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
54breq2i 3861 . . . . . . . . . 10  |-  ( l 
<Q  1Q  <->  l  <Q  [ <. 1o ,  1o >. ]  ~Q  )
65abbii 2204 . . . . . . . . 9  |-  { l  |  l  <Q  1Q }  =  { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  }
74breq1i 3860 . . . . . . . . . 10  |-  ( 1Q 
<Q  u  <->  [ <. 1o ,  1o >. ]  ~Q  <Q  u
)
87abbii 2204 . . . . . . . . 9  |-  { u  |  1Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u }
96, 8opeq12i 3635 . . . . . . . 8  |-  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
103, 9eqtri 2109 . . . . . . 7  |-  1P  =  <. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
1110oveq1i 5678 . . . . . 6  |-  ( 1P 
+P.  1P )  =  (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
1211opeq1i 3633 . . . . 5  |-  <. ( 1P  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.
13 eceq1 6343 . . . . 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 7 . . . 4  |-  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R
152, 14eqtri 2109 . . 3  |-  1R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R
1615opeq1i 3633 . 2  |-  <. 1R ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.
171, 16eqtr2i 2110 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 1290   {cab 2075   <.cop 3455   class class class wbr 3853  (class class class)co 5668   1oc1o 6190   [cec 6306    ~Q ceq 6901   1Qc1q 6903    <Q cltq 6907   1Pc1p 6914    +P. cpp 6915    ~R cer 6918   0Rc0r 6920   1Rc1r 6921   1c1 7414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-xp 4460  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fv 5038  df-ov 5671  df-ec 6310  df-1nqqs 6973  df-i1p 7089  df-1r 7341  df-1 7421
This theorem is referenced by:  pitonn  7448
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