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Theorem recriota 7425
Description: Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
Assertion
Ref Expression
recriota  |-  ( N  e.  N.  ->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  = 
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
Distinct variable group:    N, l, r, u

Proof of Theorem recriota
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 pitore 7387 . . 3  |-  ( N  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR )
2 pitoregt0 7386 . . 3  |-  ( N  e.  N.  ->  0  <RR 
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
3 axprecex 7415 . . 3  |-  ( (
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR  /\  0  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  ->  E. y  e.  RR  ( 0  <RR  y  /\  ( <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) )
41, 2, 3syl2anc 403 . 2  |-  ( N  e.  N.  ->  E. y  e.  RR  ( 0  <RR  y  /\  ( <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) )
5 simprrr 507 . . . 4  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 )
6 simprl 498 . . . . 5  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  y  e.  RR )
71adantr 270 . . . . . 6  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR )
82adantr 270 . . . . . 6  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  0  <RR  <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
9 rereceu 7424 . . . . . 6  |-  ( (
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR  /\  0  <RR  <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  ->  E! r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )
107, 8, 9syl2anc 403 . . . . 5  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  E! r  e.  RR  ( <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )
11 oveq2 5660 . . . . . . 7  |-  ( r  =  y  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  ( <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y ) )
1211eqeq1d 2096 . . . . . 6  |-  ( r  =  y  ->  (
( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1  <->  ( <. [
<. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) )
1312riota2 5630 . . . . 5  |-  ( ( y  e.  RR  /\  E! r  e.  RR  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  -> 
( ( <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1  <->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  =  y ) )
146, 10, 13syl2anc 403 . . . 4  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1  <->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  =  y ) )
155, 14mpbid 145 . . 3  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  =  y )
165oveq2d 5668 . . . 4  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y ) )  =  ( <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  1
) )
17 axresscn 7397 . . . . . . . . . 10  |-  RR  C_  CC
1817, 7sseldi 3023 . . . . . . . . 9  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  CC )
19 recnnre 7388 . . . . . . . . . . 11  |-  ( N  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR )
2019adantr 270 . . . . . . . . . 10  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR )
2117, 20sseldi 3023 . . . . . . . . 9  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  CC )
22 axmulcom 7406 . . . . . . . . 9  |-  ( (
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  CC  /\  <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  CC )  ->  ( <. [ <. (
<. { 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  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
2318, 21, 22syl2anc 403 . . . . . . . 8  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { 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  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
24 recidpirq 7395 . . . . . . . . 9  |-  ( 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 )
2524adantr 270 . . . . . . . 8  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { 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 )
2623, 25eqtr3d 2122 . . . . . . 7  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  1 )
2726oveq1d 5667 . . . . . 6  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( ( <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  x.  y
)  =  ( 1  x.  y ) )
2817, 6sseldi 3023 . . . . . . 7  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  y  e.  CC )
29 axmulass 7408 . . . . . . 7  |-  ( (
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  CC  /\ 
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  CC  /\  y  e.  CC )  ->  ( ( <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  x.  y
)  =  ( <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y ) ) )
3021, 18, 28, 29syl3anc 1174 . . . . . 6  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( ( <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  x.  y
)  =  ( <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y ) ) )
31 ax1cn 7398 . . . . . . 7  |-  1  e.  CC
32 axmulcom 7406 . . . . . . 7  |-  ( ( 1  e.  CC  /\  y  e.  CC )  ->  ( 1  x.  y
)  =  ( y  x.  1 ) )
3331, 28, 32sylancr 405 . . . . . 6  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( 1  x.  y )  =  ( y  x.  1 ) )
3427, 30, 333eqtr3d 2128 . . . . 5  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y ) )  =  ( y  x.  1 ) )
35 ax1rid 7412 . . . . . 6  |-  ( y  e.  RR  ->  (
y  x.  1 )  =  y )
366, 35syl 14 . . . . 5  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( y  x.  1 )  =  y )
3734, 36eqtrd 2120 . . . 4  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y ) )  =  y )
38 ax1rid 7412 . . . . 5  |-  ( <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR  ->  ( <. [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  1
)  =  <. [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
3920, 38syl 14 . . . 4  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( <. [
<. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  1 )  =  <. [ <. ( <. { l  |  l 
<Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
4016, 37, 393eqtr3d 2128 . . 3  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  y  =  <. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
4115, 40eqtrd 2120 . 2  |-  ( ( N  e.  N.  /\  ( y  e.  RR  /\  ( 0  <RR  y  /\  ( <. [ <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  y )  =  1 ) ) )  ->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  = 
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
424, 41rexlimddv 2493 1  |-  ( N  e.  N.  ->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  = 
<. [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   E!wreu 2361   <.cop 3449   class class class wbr 3845   ` cfv 5015   iota_crio 5607  (class class class)co 5652   1oc1o 6174   [cec 6290   N.cnpi 6831    ~Q ceq 6838   *Qcrq 6843    <Q cltq 6844   1Pc1p 6851    +P. cpp 6852    ~R cer 6855   0Rc0r 6857   CCcc 7348   RRcr 7349   0cc0 7350   1c1 7351    <RR cltrr 7354    x. cmul 7355
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-i1p 7026  df-iplp 7027  df-imp 7028  df-iltp 7029  df-enr 7272  df-nr 7273  df-plr 7274  df-mr 7275  df-ltr 7276  df-0r 7277  df-1r 7278  df-m1r 7279  df-c 7356  df-0 7357  df-1 7358  df-r 7360  df-mul 7362  df-lt 7363
This theorem is referenced by:  axcaucvglemcau  7433
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