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Theorem ruclem6 14364
Description: Lemma for ruc 14372. 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 5880 . . . . . 6  |-  ( G `
 0 )  =  (  seq 0 ( D ,  C ) `
 0 )
3 0z 10972 . . . . . . 7  |-  0  e.  ZZ
4 seq1 12264 . . . . . . 7  |-  ( 0  e.  ZZ  ->  (  seq 0 ( D ,  C ) `  0
)  =  ( C `
 0 ) )
53, 4ax-mp 5 . . . . . 6  |-  (  seq 0 ( D ,  C ) `  0
)  =  ( C `
 0 )
62, 5eqtri 2493 . . . . 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 14363 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
116, 10syl5eqr 2519 . . . 4  |-  ( ph  ->  ( C `  0
)  =  <. 0 ,  1 >. )
12 0re 9661 . . . . 5  |-  0  e.  RR
13 1re 9660 . . . . 5  |-  1  e.  RR
14 opelxpi 4871 . . . . 5  |-  ( ( 0  e.  RR  /\  1  e.  RR )  -> 
<. 0 ,  1
>.  e.  ( RR  X.  RR ) )
1512, 13, 14mp2an 686 . . . 4  |-  <. 0 ,  1 >.  e.  ( RR  X.  RR )
1611, 15syl6eqel 2557 . . 3  |-  ( ph  ->  ( C `  0
)  e.  ( RR 
X.  RR ) )
17 1st2nd2 6849 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1817ad2antrl 742 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1918oveq1d 6323 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. D w ) )
207adantr 472 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  F : NN --> RR )
218adantr 472 . . . . . 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 6842 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
2322ad2antrl 742 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 1st `  z )  e.  RR )
24 xp2nd 6843 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
2524ad2antrl 742 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 2nd `  z )  e.  RR )
26 simprr 774 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  w  e.  RR )
27 eqid 2471 . . . . . 6  |-  ( 1st `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 1st `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
28 eqid 2471 . . . . . 6  |-  ( 2nd `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 2nd `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
2920, 21, 23, 25, 26, 27, 28ruclem1 14360 . . . . 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 1042 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w )  e.  ( RR 
X.  RR ) )
3119, 30eqeltrd 2549 . . 3  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  e.  ( RR  X.  RR ) )
32 nn0uz 11217 . . 3  |-  NN0  =  ( ZZ>= `  0 )
33 0zd 10973 . . 3  |-  ( ph  ->  0  e.  ZZ )
34 0p1e1 10743 . . . . . . 7  |-  ( 0  +  1 )  =  1
3534fveq2i 5882 . . . . . 6  |-  ( ZZ>= `  ( 0  +  1 ) )  =  (
ZZ>= `  1 )
36 nnuz 11218 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3735, 36eqtr4i 2496 . . . . 5  |-  ( ZZ>= `  ( 0  +  1 ) )  =  NN
3837eleq2i 2541 . . . 4  |-  ( z  e.  ( ZZ>= `  (
0  +  1 ) )  <->  z  e.  NN )
399equncomi 3571 . . . . . . . 8  |-  C  =  ( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } )
4039fveq1i 5880 . . . . . . 7  |-  ( C `
 z )  =  ( ( F  u.  {
<. 0 ,  <. 0 ,  1 >. >. } ) `  z
)
41 nnne0 10664 . . . . . . . . 9  |-  ( z  e.  NN  ->  z  =/=  0 )
4241necomd 2698 . . . . . . . 8  |-  ( z  e.  NN  ->  0  =/=  z )
43 fvunsn 6112 . . . . . . . 8  |-  ( 0  =/=  z  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4442, 43syl 17 . . . . . . 7  |-  ( z  e.  NN  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4540, 44syl5eq 2517 . . . . . 6  |-  ( z  e.  NN  ->  ( C `  z )  =  ( F `  z ) )
4645adantl 473 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  =  ( F `  z
) )
477ffvelrnda 6037 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( F `
 z )  e.  RR )
4846, 47eqeltrd 2549 . . . 4  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  e.  RR )
4938, 48sylan2b 483 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( C `  z )  e.  RR )
5016, 31, 32, 33, 49seqf2 12270 . 2  |-  ( ph  ->  seq 0 ( D ,  C ) : NN0 --> ( RR  X.  RR ) )
511feq1i 5730 . 2  |-  ( G : NN0 --> ( RR 
X.  RR )  <->  seq 0
( D ,  C
) : NN0 --> ( RR 
X.  RR ) )
5250, 51sylibr 217 1  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   [_csb 3349    u. cun 3388   ifcif 3872   {csn 3959   <.cop 3965   class class class wbr 4395    X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182    seqcseq 12251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-seq 12252
This theorem is referenced by:  ruclem8  14366  ruclem9  14367  ruclem10  14368  ruclem11  14369  ruclem12  14370
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