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Theorem ruclem6 12824
Description: Lemma for ruc 12832. 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 5721 . . . . . 6  |-  ( G `
 0 )  =  (  seq  0 ( D ,  C ) `
 0 )
3 0z 10283 . . . . . . 7  |-  0  e.  ZZ
4 seq1 11326 . . . . . . 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 2455 . . . . 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 12823 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
116, 10syl5eqr 2481 . . . 4  |-  ( ph  ->  ( C `  0
)  =  <. 0 ,  1 >. )
12 0re 9081 . . . . 5  |-  0  e.  RR
13 1re 9080 . . . . 5  |-  1  e.  RR
14 opelxpi 4902 . . . . 5  |-  ( ( 0  e.  RR  /\  1  e.  RR )  -> 
<. 0 ,  1
>.  e.  ( RR  X.  RR ) )
1512, 13, 14mp2an 654 . . . 4  |-  <. 0 ,  1 >.  e.  ( RR  X.  RR )
1611, 15syl6eqel 2523 . . 3  |-  ( ph  ->  ( C `  0
)  e.  ( RR 
X.  RR ) )
17 1st2nd2 6378 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1817ad2antrl 709 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1918oveq1d 6088 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. D w ) )
207adantr 452 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  F : NN --> RR )
218adantr 452 . . . . . 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 6368 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
2322ad2antrl 709 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 1st `  z )  e.  RR )
24 xp2nd 6369 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
2524ad2antrl 709 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 2nd `  z )  e.  RR )
26 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  w  e.  RR )
27 eqid 2435 . . . . . 6  |-  ( 1st `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 1st `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
28 eqid 2435 . . . . . 6  |-  ( 2nd `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 2nd `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
2920, 21, 23, 25, 26, 27, 28ruclem1 12820 . . . . 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 969 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w )  e.  ( RR 
X.  RR ) )
3119, 30eqeltrd 2509 . . 3  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  e.  ( RR  X.  RR ) )
32 nn0uz 10510 . . 3  |-  NN0  =  ( ZZ>= `  0 )
333a1i 11 . . 3  |-  ( ph  ->  0  e.  ZZ )
34 0p1e1 10083 . . . . . . 7  |-  ( 0  +  1 )  =  1
3534fveq2i 5723 . . . . . 6  |-  ( ZZ>= `  ( 0  +  1 ) )  =  (
ZZ>= `  1 )
36 nnuz 10511 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3735, 36eqtr4i 2458 . . . . 5  |-  ( ZZ>= `  ( 0  +  1 ) )  =  NN
3837eleq2i 2499 . . . 4  |-  ( z  e.  ( ZZ>= `  (
0  +  1 ) )  <->  z  e.  NN )
399equncomi 3485 . . . . . . . 8  |-  C  =  ( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } )
4039fveq1i 5721 . . . . . . 7  |-  ( C `
 z )  =  ( ( F  u.  {
<. 0 ,  <. 0 ,  1 >. >. } ) `  z
)
41 nnne0 10022 . . . . . . . . 9  |-  ( z  e.  NN  ->  z  =/=  0 )
4241necomd 2681 . . . . . . . 8  |-  ( z  e.  NN  ->  0  =/=  z )
43 fvunsn 5917 . . . . . . . 8  |-  ( 0  =/=  z  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4442, 43syl 16 . . . . . . 7  |-  ( z  e.  NN  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4540, 44syl5eq 2479 . . . . . 6  |-  ( z  e.  NN  ->  ( C `  z )  =  ( F `  z ) )
4645adantl 453 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  =  ( F `  z
) )
477ffvelrnda 5862 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( F `
 z )  e.  RR )
4846, 47eqeltrd 2509 . . . 4  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  e.  RR )
4938, 48sylan2b 462 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( C `  z )  e.  RR )
5016, 31, 32, 33, 49seqf2 11332 . 2  |-  ( ph  ->  seq  0 ( D ,  C ) : NN0 --> ( RR  X.  RR ) )
511feq1i 5577 . 2  |-  ( G : NN0 --> ( RR 
X.  RR )  <->  seq  0
( D ,  C
) : NN0 --> ( RR 
X.  RR ) )
5250, 51sylibr 204 1  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   [_csb 3243    u. cun 3310   ifcif 3731   {csn 3806   <.cop 3809   class class class wbr 4204    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   RRcr 8979   0cc0 8980   1c1 8981    + caddc 8983    < clt 9110    / cdiv 9667   NNcn 9990   2c2 10039   NN0cn0 10211   ZZcz 10272   ZZ>=cuz 10478    seq cseq 11313
This theorem is referenced by:  ruclem8  12826  ruclem9  12827  ruclem10  12828  ruclem11  12829  ruclem12  12830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-seq 11314
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