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Theorem recidpirq 7298
Description: A real number times its reciprocal is one, where reciprocal is expressed with  *Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
Assertion
Ref Expression
recidpirq  |-  ( N  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  1 )
Distinct variable group:    N, l, u

Proof of Theorem recidpirq
StepHypRef Expression
1 nnprlu 7015 . . . 4  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
2 prsrcl 7232 . . . 4  |-  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P.  ->  [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
31, 2syl 14 . . 3  |-  ( N  e.  N.  ->  [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
4 recnnpr 7010 . . . 4  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
5 prsrcl 7232 . . . 4  |-  ( <. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P.  ->  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
64, 5syl 14 . . 3  |-  ( N  e.  N.  ->  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
7 mulresr 7278 . . 3  |-  ( ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R.  /\  [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  R. )  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  .R  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
83, 6, 7syl2anc 403 . 2  |-  ( N  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  .R  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >. )
9 1pr 7016 . . . . . . . 8  |-  1P  e.  P.
109a1i 9 . . . . . . 7  |-  ( N  e.  N.  ->  1P  e.  P. )
11 addclpr 6999 . . . . . . 7  |-  ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P.  /\  1P  e.  P. )  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
121, 10, 11syl2anc 403 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
13 addclpr 6999 . . . . . . 7  |-  ( (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P.  /\  1P  e.  P. )  ->  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  e.  P. )
144, 10, 13syl2anc 403 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  e.  P. )
15 mulsrpr 7195 . . . . . 6  |-  ( ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e.  P.  /\  1P  e.  P. )  /\  (
( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  .R  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. ( ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  )
1612, 10, 14, 10, 15syl22anc 1171 . . . . 5  |-  ( N  e.  N.  ->  ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  .R  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. ( ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  )
17 recidpipr 7296 . . . . . . 7  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  .P.  <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >. )  =  1P )
181, 4, 17recidpirqlemcalc 7297 . . . . . 6  |-  ( N  e.  N.  ->  (
( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) )  +P. 
1P )  =  ( ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) )
19 df-1r 7181 . . . . . . . 8  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
2019eqeq2i 2093 . . . . . . 7  |-  ( [
<. ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  =  1R  <->  [ <. ( ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
21 mulclpr 7034 . . . . . . . . . 10  |-  ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P.  /\  ( <. { l  |  l  <Q 
( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  e.  P. )  ->  ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  e.  P. )
2212, 14, 21syl2anc 403 . . . . . . . . 9  |-  ( N  e.  N.  ->  (
( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  e.  P. )
239, 9pm3.2i 266 . . . . . . . . . 10  |-  ( 1P  e.  P.  /\  1P  e.  P. )
24 mulclpr 7034 . . . . . . . . . 10  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  .P.  1P )  e.  P. )
2523, 24mp1i 10 . . . . . . . . 9  |-  ( N  e.  N.  ->  ( 1P  .P.  1P )  e. 
P. )
26 addclpr 6999 . . . . . . . . 9  |-  ( ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  e.  P.  /\  ( 1P  .P.  1P )  e.  P. )  -> 
( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) )  e. 
P. )
2722, 25, 26syl2anc 403 . . . . . . . 8  |-  ( N  e.  N.  ->  (
( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) )  e. 
P. )
28 mulclpr 7034 . . . . . . . . . 10  |-  ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P.  /\  1P  e.  P. )  ->  ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  e.  P. )
2912, 10, 28syl2anc 403 . . . . . . . . 9  |-  ( N  e.  N.  ->  (
( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  e.  P. )
30 mulclpr 7034 . . . . . . . . . 10  |-  ( ( 1P  e.  P.  /\  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  e.  P. )  ->  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) )  e.  P. )
3110, 14, 30syl2anc 403 . . . . . . . . 9  |-  ( N  e.  N.  ->  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) )  e.  P. )
32 addclpr 6999 . . . . . . . . 9  |-  ( ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  1P )  e.  P.  /\  ( 1P  .P.  ( <. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) )  e.  P. )  ->  ( ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) )  e.  P. )
3329, 31, 32syl2anc 403 . . . . . . . 8  |-  ( N  e.  N.  ->  (
( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) )  e.  P. )
34 addclpr 6999 . . . . . . . . 9  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
3523, 34mp1i 10 . . . . . . . 8  |-  ( N  e.  N.  ->  ( 1P  +P.  1P )  e. 
P. )
36 enreceq 7185 . . . . . . . 8  |-  ( ( ( ( ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) )  e. 
P.  /\  ( (
( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) )  e.  P. )  /\  ( ( 1P 
+P.  1P )  e.  P.  /\  1P  e.  P. )
)  ->  ( [ <. ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) )  +P. 
1P )  =  ( ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) ) )
3727, 33, 35, 10, 36syl22anc 1171 . . . . . . 7  |-  ( N  e.  N.  ->  ( [ <. ( ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) )  +P. 
1P )  =  ( ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) ) )
3820, 37syl5bb 190 . . . . . 6  |-  ( N  e.  N.  ->  ( [ <. ( ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  =  1R  <->  ( ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) )  +P. 
1P )  =  ( ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) ) )
3918, 38mpbird 165 . . . . 5  |-  ( N  e.  N.  ->  [ <. ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )
)  +P.  ( 1P  .P.  1P ) ) ,  ( ( ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ) ) >. ]  ~R  =  1R )
4016, 39eqtrd 2115 . . . 4  |-  ( N  e.  N.  ->  ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  .R  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  =  1R )
4140opeq1d 3602 . . 3  |-  ( N  e.  N.  ->  <. ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  .R  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >.  =  <. 1R ,  0R >. )
42 df-1 7261 . . 3  |-  1  =  <. 1R ,  0R >.
4341, 42syl6eqr 2133 . 2  |-  ( N  e.  N.  ->  <. ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  .R  [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ,  0R >.  =  1
)
448, 43eqtrd 2115 1  |-  ( N  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2069   <.cop 3425   class class class wbr 3811   ` cfv 4969  (class class class)co 5591   1oc1o 6106   [cec 6220   N.cnpi 6734    ~Q ceq 6741   *Qcrq 6746    <Q cltq 6747   P.cnp 6753   1Pc1p 6754    +P. cpp 6755    .P. cmp 6756    ~R cer 6758   R.cnr 6759   0Rc0r 6760   1Rc1r 6761    .R cmr 6764   1c1 7254    x. cmul 7258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-i1p 6929  df-iplp 6930  df-imp 6931  df-enr 7175  df-nr 7176  df-plr 7177  df-mr 7178  df-0r 7180  df-1r 7181  df-m1r 7182  df-c 7259  df-1 7261  df-mul 7265
This theorem is referenced by:  recriota  7328
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