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Theorem caucvgsrlemfv 7819
Description: Lemma for caucvgsr 7830. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
Hypotheses
Ref Expression
caucvgsr.f  |-  ( ph  ->  F : N. --> R. )
caucvgsr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
caucvgsrlemgt1.gt1  |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )
caucvgsrlemf.xfr  |-  G  =  ( x  e.  N.  |->  ( iota_ y  e.  P.  ( F `  x )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
Assertion
Ref Expression
caucvgsrlemfv  |-  ( (
ph  /\  A  e.  N. )  ->  [ <. ( ( G `  A
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )
Distinct variable groups:    A, m    x, A, y    m, F    x, F, y    ph, x
Allowed substitution hints:    ph( y, u, k, m, n, l)    A( u, k, n, l)    F( u, k, n, l)    G( x, y, u, k, m, n, l)

Proof of Theorem caucvgsrlemfv
StepHypRef Expression
1 caucvgsrlemf.xfr . . . . . . 7  |-  G  =  ( x  e.  N.  |->  ( iota_ y  e.  P.  ( F `  x )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
21a1i 9 . . . . . 6  |-  ( (
ph  /\  A  e.  N. )  ->  G  =  ( x  e.  N.  |->  ( iota_ y  e.  P.  ( F `  x )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
) )
3 fveq2 5534 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
43eqeq1d 2198 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  =  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  <->  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
54riotabidv 5853 . . . . . . 7  |-  ( x  =  A  ->  ( iota_ y  e.  P.  ( F `  x )  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )  =  ( iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
65adantl 277 . . . . . 6  |-  ( ( ( ph  /\  A  e.  N. )  /\  x  =  A )  ->  ( iota_ y  e.  P.  ( F `  x )  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )  =  ( iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
7 simpr 110 . . . . . 6  |-  ( (
ph  /\  A  e.  N. )  ->  A  e. 
N. )
8 caucvgsr.f . . . . . . 7  |-  ( ph  ->  F : N. --> R. )
9 caucvgsrlemgt1.gt1 . . . . . . 7  |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )
108, 9caucvgsrlemcl 7817 . . . . . 6  |-  ( (
ph  /\  A  e.  N. )  ->  ( iota_ y  e.  P.  ( F `
 A )  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )  e. 
P. )
112, 6, 7, 10fvmptd 5617 . . . . 5  |-  ( (
ph  /\  A  e.  N. )  ->  ( G `
 A )  =  ( iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )
)
1211oveq1d 5910 . . . 4  |-  ( (
ph  /\  A  e.  N. )  ->  ( ( G `  A )  +P.  1P )  =  ( ( iota_ y  e. 
P.  ( F `  A )  =  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  )  +P.  1P ) )
1312opeq1d 3799 . . 3  |-  ( (
ph  /\  A  e.  N. )  ->  <. (
( G `  A
)  +P.  1P ) ,  1P >.  =  <. ( ( iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )  +P.  1P ) ,  1P >. )
1413eceq1d 6594 . 2  |-  ( (
ph  /\  A  e.  N. )  ->  [ <. ( ( G `  A
)  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( ( iota_ y  e.  P.  ( F `
 A )  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )  +P. 
1P ) ,  1P >. ]  ~R  )
15 eqcom 2191 . . . . . . 7  |-  ( ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  <->  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )
1615a1i 9 . . . . . 6  |-  ( y  e.  P.  ->  (
( F `  A
)  =  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  <->  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) ) )
1716riotabiia 5868 . . . . 5  |-  ( iota_ y  e.  P.  ( F `
 A )  =  [ <. ( y  +P. 
1P ) ,  1P >. ]  ~R  )  =  ( iota_ y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )
1817oveq1i 5905 . . . 4  |-  ( (
iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )  +P.  1P )  =  ( ( iota_ y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )  +P. 
1P )
1918opeq1i 3796 . . 3  |-  <. (
( iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )  +P.  1P ) ,  1P >.  =  <. ( ( iota_ y  e.  P.  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )  +P. 
1P ) ,  1P >.
20 eceq1 6593 . . 3  |-  ( <.
( ( iota_ y  e. 
P.  ( F `  A )  =  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  )  +P.  1P ) ,  1P >.  =  <. ( ( iota_ y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )  +P. 
1P ) ,  1P >.  ->  [ <. (
( iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( ( iota_ y  e. 
P.  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )  +P.  1P ) ,  1P >. ]  ~R  )
2119, 20mp1i 10 . 2  |-  ( (
ph  /\  A  e.  N. )  ->  [ <. ( ( iota_ y  e.  P.  ( F `  A )  =  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  )  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( ( iota_ y  e. 
P.  [ <. (
y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )  +P.  1P ) ,  1P >. ]  ~R  )
228ffvelcdmda 5671 . . 3  |-  ( (
ph  /\  A  e.  N. )  ->  ( F `
 A )  e. 
R. )
23 0lt1sr 7793 . . . 4  |-  0R  <R  1R
24 fveq2 5534 . . . . . . 7  |-  ( m  =  A  ->  ( F `  m )  =  ( F `  A ) )
2524breq2d 4030 . . . . . 6  |-  ( m  =  A  ->  ( 1R  <R  ( F `  m )  <->  1R  <R  ( F `  A )
) )
2625rspcv 2852 . . . . 5  |-  ( A  e.  N.  ->  ( A. m  e.  N.  1R  <R  ( F `  m )  ->  1R  <R  ( F `  A
) ) )
279, 26mpan9 281 . . . 4  |-  ( (
ph  /\  A  e.  N. )  ->  1R  <R  ( F `  A ) )
28 ltsosr 7792 . . . . 5  |-  <R  Or  R.
29 ltrelsr 7766 . . . . 5  |-  <R  C_  ( R.  X.  R. )
3028, 29sotri 5042 . . . 4  |-  ( ( 0R  <R  1R  /\  1R  <R  ( F `  A
) )  ->  0R  <R  ( F `  A
) )
3123, 27, 30sylancr 414 . . 3  |-  ( (
ph  /\  A  e.  N. )  ->  0R  <R  ( F `  A ) )
32 prsrriota 7816 . . 3  |-  ( ( ( F `  A
)  e.  R.  /\  0R  <R  ( F `  A ) )  ->  [ <. ( ( iota_ y  e.  P.  [ <. ( y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( F `  A ) )
3322, 31, 32syl2anc 411 . 2  |-  ( (
ph  /\  A  e.  N. )  ->  [ <. ( ( iota_ y  e.  P.  [
<. ( y  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( F `  A ) )
3414, 21, 333eqtrd 2226 1  |-  ( (
ph  /\  A  e.  N. )  ->  [ <. ( ( G `  A
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   <.cop 3610   class class class wbr 4018    |-> cmpt 4079   -->wf 5231   ` cfv 5235   iota_crio 5850  (class class class)co 5895   1oc1o 6433   [cec 6556   N.cnpi 7300    <N clti 7303    ~Q ceq 7307   *Qcrq 7312    <Q cltq 7313   P.cnp 7319   1Pc1p 7320    +P. cpp 7321    ~R cer 7324   R.cnr 7325   0Rc0r 7326   1Rc1r 7327    +R cplr 7329    <R cltr 7331
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-2o 6441  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-enq0 7452  df-nq0 7453  df-0nq0 7454  df-plq0 7455  df-mq0 7456  df-inp 7494  df-i1p 7495  df-iplp 7496  df-iltp 7498  df-enr 7754  df-nr 7755  df-ltr 7758  df-0r 7759  df-1r 7760
This theorem is referenced by:  caucvgsrlemcau  7821  caucvgsrlembound  7822  caucvgsrlemgt1  7823
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