MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruclem6 Unicode version

Theorem ruclem6 12513
Description: Lemma for ruc 12521. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1  |-  ( ph  ->  F : NN --> RR )
ruc.2  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
ruc.4  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
ruc.5  |-  G  =  seq  0 ( D ,  C )
Assertion
Ref Expression
ruclem6  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
Distinct variable groups:    x, m, y, F    m, G, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem6
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.5 . . . . . . 7  |-  G  =  seq  0 ( D ,  C )
21fveq1i 5526 . . . . . 6  |-  ( G `
 0 )  =  (  seq  0 ( D ,  C ) `
 0 )
3 0z 10035 . . . . . . 7  |-  0  e.  ZZ
4 seq1 11059 . . . . . . 7  |-  ( 0  e.  ZZ  ->  (  seq  0 ( D ,  C ) `  0
)  =  ( C `
 0 ) )
53, 4ax-mp 8 . . . . . 6  |-  (  seq  0 ( D ,  C ) `  0
)  =  ( C `
 0 )
62, 5eqtri 2303 . . . . 5  |-  ( G `
 0 )  =  ( C `  0
)
7 ruc.1 . . . . . 6  |-  ( ph  ->  F : NN --> RR )
8 ruc.2 . . . . . 6  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
9 ruc.4 . . . . . 6  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
107, 8, 9, 1ruclem4 12512 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
116, 10syl5eqr 2329 . . . 4  |-  ( ph  ->  ( C `  0
)  =  <. 0 ,  1 >. )
12 0re 8838 . . . . 5  |-  0  e.  RR
13 1re 8837 . . . . 5  |-  1  e.  RR
14 opelxpi 4721 . . . . 5  |-  ( ( 0  e.  RR  /\  1  e.  RR )  -> 
<. 0 ,  1
>.  e.  ( RR  X.  RR ) )
1512, 13, 14mp2an 653 . . . 4  |-  <. 0 ,  1 >.  e.  ( RR  X.  RR )
1611, 15syl6eqel 2371 . . 3  |-  ( ph  ->  ( C `  0
)  e.  ( RR 
X.  RR ) )
17 1st2nd2 6159 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1817ad2antrl 708 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1918oveq1d 5873 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. D w ) )
207adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  F : NN --> RR )
218adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
22 xp1st 6149 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
2322ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 1st `  z )  e.  RR )
24 xp2nd 6150 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
2524ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 2nd `  z )  e.  RR )
26 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  w  e.  RR )
27 eqid 2283 . . . . . 6  |-  ( 1st `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 1st `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
28 eqid 2283 . . . . . 6  |-  ( 2nd `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 2nd `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
2920, 21, 23, 25, 26, 27, 28ruclem1 12509 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w )  e.  ( RR 
X.  RR )  /\  ( 1st `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  if ( ( ( ( 1st `  z )  +  ( 2nd `  z
) )  /  2
)  <  w , 
( 1st `  z
) ,  ( ( ( ( ( 1st `  z )  +  ( 2nd `  z ) )  /  2 )  +  ( 2nd `  z
) )  /  2
) )  /\  ( 2nd `  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. D w ) )  =  if ( ( ( ( 1st `  z )  +  ( 2nd `  z ) )  /  2 )  <  w ,  ( ( ( 1st `  z
)  +  ( 2nd `  z ) )  / 
2 ) ,  ( 2nd `  z ) ) ) )
3029simp1d 967 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w )  e.  ( RR 
X.  RR ) )
3119, 30eqeltrd 2357 . . 3  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  e.  ( RR  X.  RR ) )
32 nn0uz 10262 . . 3  |-  NN0  =  ( ZZ>= `  0 )
333a1i 10 . . 3  |-  ( ph  ->  0  e.  ZZ )
34 0p1e1 9839 . . . . . . 7  |-  ( 0  +  1 )  =  1
3534fveq2i 5528 . . . . . 6  |-  ( ZZ>= `  ( 0  +  1 ) )  =  (
ZZ>= `  1 )
36 nnuz 10263 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3735, 36eqtr4i 2306 . . . . 5  |-  ( ZZ>= `  ( 0  +  1 ) )  =  NN
3837eleq2i 2347 . . . 4  |-  ( z  e.  ( ZZ>= `  (
0  +  1 ) )  <->  z  e.  NN )
399equncomi 3321 . . . . . . . 8  |-  C  =  ( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } )
4039fveq1i 5526 . . . . . . 7  |-  ( C `
 z )  =  ( ( F  u.  {
<. 0 ,  <. 0 ,  1 >. >. } ) `  z
)
41 nnne0 9778 . . . . . . . . 9  |-  ( z  e.  NN  ->  z  =/=  0 )
4241necomd 2529 . . . . . . . 8  |-  ( z  e.  NN  ->  0  =/=  z )
43 fvunsn 5712 . . . . . . . 8  |-  ( 0  =/=  z  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4442, 43syl 15 . . . . . . 7  |-  ( z  e.  NN  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4540, 44syl5eq 2327 . . . . . 6  |-  ( z  e.  NN  ->  ( C `  z )  =  ( F `  z ) )
4645adantl 452 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  =  ( F `  z
) )
47 ffvelrn 5663 . . . . . 6  |-  ( ( F : NN --> RR  /\  z  e.  NN )  ->  ( F `  z
)  e.  RR )
487, 47sylan 457 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( F `
 z )  e.  RR )
4946, 48eqeltrd 2357 . . . 4  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  e.  RR )
5038, 49sylan2b 461 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( C `  z )  e.  RR )
5116, 31, 32, 33, 50seqf2 11065 . 2  |-  ( ph  ->  seq  0 ( D ,  C ) : NN0 --> ( RR  X.  RR ) )
521feq1i 5383 . 2  |-  ( G : NN0 --> ( RR 
X.  RR )  <->  seq  0
( D ,  C
) : NN0 --> ( RR 
X.  RR ) )
5351, 52sylibr 203 1  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   [_csb 3081    u. cun 3150   ifcif 3565   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046
This theorem is referenced by:  ruclem8  12515  ruclem9  12516  ruclem10  12517  ruclem11  12518  ruclem12  12519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
  Copyright terms: Public domain W3C validator