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Theorem pitonnlem1 7844
Description: Lemma for pitonn 7847. 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 7819 . 2  |-  1  =  <. 1R ,  0R >.
2 df-1r 7731 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
3 df-i1p 7466 . . . . . . . 8  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
4 df-1nqqs 7350 . . . . . . . . . . 11  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
54breq2i 4012 . . . . . . . . . 10  |-  ( l 
<Q  1Q  <->  l  <Q  [ <. 1o ,  1o >. ]  ~Q  )
65abbii 2293 . . . . . . . . 9  |-  { l  |  l  <Q  1Q }  =  { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  }
74breq1i 4011 . . . . . . . . . 10  |-  ( 1Q 
<Q  u  <->  [ <. 1o ,  1o >. ]  ~Q  <Q  u
)
87abbii 2293 . . . . . . . . 9  |-  { u  |  1Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u }
96, 8opeq12i 3784 . . . . . . . 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 5885 . . . . . 6  |-  ( 1P 
+P.  1P )  =  (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
1211opeq1i 3782 . . . . 5  |-  <. ( 1P  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.
13 eceq1 6570 . . . . 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 3782 . 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 3596   class class class wbr 4004  (class class class)co 5875   1oc1o 6410   [cec 6533    ~Q ceq 7278   1Qc1q 7280    <Q cltq 7284   1Pc1p 7291    +P. cpp 7292    ~R cer 7295   0Rc0r 7297   1Rc1r 7298   1c1 7812
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 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fv 5225  df-ov 5878  df-ec 6537  df-1nqqs 7350  df-i1p 7466  df-1r 7731  df-1 7819
This theorem is referenced by:  pitonn  7847
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