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Theorem recidpirq 7634
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 7329 . . . 4  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
2 prsrcl 7560 . . . 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 7324 . . . 4  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
5 prsrcl 7560 . . . 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 7614 . . 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 408 . 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 7330 . . . . . . . 8  |-  1P  e.  P.
109a1i 9 . . . . . . 7  |-  ( N  e.  N.  ->  1P  e.  P. )
11 addclpr 7313 . . . . . . 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 408 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
13 addclpr 7313 . . . . . . 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 408 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P )  e.  P. )
15 mulsrpr 7522 . . . . . 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 1202 . . . . 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 7632 . . . . . . 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 7633 . . . . . 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 7508 . . . . . . . 8  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
2019eqeq2i 2128 . . . . . . 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 7348 . . . . . . . . . 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 408 . . . . . . . . 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 270 . . . . . . . . . 10  |-  ( 1P  e.  P.  /\  1P  e.  P. )
24 mulclpr 7348 . . . . . . . . . 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 7313 . . . . . . . . 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 408 . . . . . . . 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 7348 . . . . . . . . . 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 408 . . . . . . . . 9  |-  ( N  e.  N.  ->  (
( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  .P. 
1P )  e.  P. )
30 mulclpr 7348 . . . . . . . . . 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 408 . . . . . . . . 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 7313 . . . . . . . . 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 408 . . . . . . . 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 7313 . . . . . . . . 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 7512 . . . . . . . 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 1202 . . . . . . 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 191 . . . . . 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 166 . . . . 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 2150 . . . 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 3681 . . 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 7596 . . 3  |-  1  =  <. 1R ,  0R >.
4341, 42syl6eqr 2168 . 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 2150 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1oc1o 6274   [cec 6395   N.cnpi 7048    ~Q ceq 7055   *Qcrq 7060    <Q cltq 7061   P.cnp 7067   1Pc1p 7068    +P. cpp 7069    .P. cmp 7070    ~R cer 7072   R.cnr 7073   0Rc0r 7074   1Rc1r 7075    .R cmr 7078   1c1 7589    x. cmul 7593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-iplp 7244  df-imp 7245  df-enr 7502  df-nr 7503  df-plr 7504  df-mr 7505  df-0r 7507  df-1r 7508  df-m1r 7509  df-c 7594  df-1 7596  df-mul 7600
This theorem is referenced by:  recriota  7666
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