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Theorem pitonnlem1 7846
Description: Lemma for pitonn 7849. 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 7821 . 2  |-  1  =  <. 1R ,  0R >.
2 df-1r 7733 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
3 df-i1p 7468 . . . . . . . 8  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
4 df-1nqqs 7352 . . . . . . . . . . 11  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
54breq2i 4013 . . . . . . . . . 10  |-  ( l 
<Q  1Q  <->  l  <Q  [ <. 1o ,  1o >. ]  ~Q  )
65abbii 2293 . . . . . . . . 9  |-  { l  |  l  <Q  1Q }  =  { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  }
74breq1i 4012 . . . . . . . . . 10  |-  ( 1Q 
<Q  u  <->  [ <. 1o ,  1o >. ]  ~Q  <Q  u
)
87abbii 2293 . . . . . . . . 9  |-  { u  |  1Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u }
96, 8opeq12i 3785 . . . . . . . 8  |-  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
103, 9eqtri 2198 . . . . . . 7  |-  1P  =  <. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
1110oveq1i 5887 . . . . . 6  |-  ( 1P 
+P.  1P )  =  (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
1211opeq1i 3783 . . . . 5  |-  <. ( 1P  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.
13 eceq1 6572 . . . . 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 2198 . . 3  |-  1R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R
1615opeq1i 3783 . 2  |-  <. 1R ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.
171, 16eqtr2i 2199 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 1353   {cab 2163   <.cop 3597   class class class wbr 4005  (class class class)co 5877   1oc1o 6412   [cec 6535    ~Q ceq 7280   1Qc1q 7282    <Q cltq 7286   1Pc1p 7293    +P. cpp 7294    ~R cer 7297   0Rc0r 7299   1Rc1r 7300   1c1 7814
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fv 5226  df-ov 5880  df-ec 6539  df-1nqqs 7352  df-i1p 7468  df-1r 7733  df-1 7821
This theorem is referenced by:  pitonn  7849
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