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Theorem pitonnlem1 6979
Description: Lemma for pitonn 6982. 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 6955 . 2  |-  1  =  <. 1R ,  0R >.
2 df-1r 6875 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
3 df-i1p 6623 . . . . . . . 8  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
4 df-1nqqs 6507 . . . . . . . . . . 11  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
54breq2i 3800 . . . . . . . . . 10  |-  ( l 
<Q  1Q  <->  l  <Q  [ <. 1o ,  1o >. ]  ~Q  )
65abbii 2169 . . . . . . . . 9  |-  { l  |  l  <Q  1Q }  =  { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  }
74breq1i 3799 . . . . . . . . . 10  |-  ( 1Q 
<Q  u  <->  [ <. 1o ,  1o >. ]  ~Q  <Q  u
)
87abbii 2169 . . . . . . . . 9  |-  { u  |  1Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u }
96, 8opeq12i 3582 . . . . . . . 8  |-  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
103, 9eqtri 2076 . . . . . . 7  |-  1P  =  <. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.
1110oveq1i 5550 . . . . . 6  |-  ( 1P 
+P.  1P )  =  (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
1211opeq1i 3580 . . . . 5  |-  <. ( 1P  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.
13 eceq1 6172 . . . . 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 2076 . . 3  |-  1R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R
1615opeq1i 3580 . 2  |-  <. 1R ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.
171, 16eqtr2i 2077 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 1259   {cab 2042   <.cop 3406   class class class wbr 3792  (class class class)co 5540   1oc1o 6025   [cec 6135    ~Q ceq 6435   1Qc1q 6437    <Q cltq 6441   1Pc1p 6448    +P. cpp 6449    ~R cer 6452   0Rc0r 6454   1Rc1r 6455   1c1 6948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fv 4938  df-ov 5543  df-ec 6139  df-1nqqs 6507  df-i1p 6623  df-1r 6875  df-1 6955
This theorem is referenced by:  pitonn  6982
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