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Theorem ruclem6 13605
Description: Lemma for ruc 13613. 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 5776 . . . . . 6  |-  ( G `
 0 )  =  (  seq 0 ( D ,  C ) `
 0 )
3 0z 10744 . . . . . . 7  |-  0  e.  ZZ
4 seq1 11906 . . . . . . 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 2478 . . . . 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 13604 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
116, 10syl5eqr 2504 . . . 4  |-  ( ph  ->  ( C `  0
)  =  <. 0 ,  1 >. )
12 0re 9473 . . . . 5  |-  0  e.  RR
13 1re 9472 . . . . 5  |-  1  e.  RR
14 opelxpi 4955 . . . . 5  |-  ( ( 0  e.  RR  /\  1  e.  RR )  -> 
<. 0 ,  1
>.  e.  ( RR  X.  RR ) )
1512, 13, 14mp2an 672 . . . 4  |-  <. 0 ,  1 >.  e.  ( RR  X.  RR )
1611, 15syl6eqel 2544 . . 3  |-  ( ph  ->  ( C `  0
)  e.  ( RR 
X.  RR ) )
17 1st2nd2 6699 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1817ad2antrl 727 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1918oveq1d 6191 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. D w ) )
207adantr 465 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  F : NN --> RR )
218adantr 465 . . . . . 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 6692 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
2322ad2antrl 727 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 1st `  z )  e.  RR )
24 xp2nd 6693 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
2524ad2antrl 727 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 2nd `  z )  e.  RR )
26 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  w  e.  RR )
27 eqid 2450 . . . . . 6  |-  ( 1st `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 1st `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
28 eqid 2450 . . . . . 6  |-  ( 2nd `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 2nd `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
2920, 21, 23, 25, 26, 27, 28ruclem1 13601 . . . . 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 1000 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w )  e.  ( RR 
X.  RR ) )
3119, 30eqeltrd 2536 . . 3  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  e.  ( RR  X.  RR ) )
32 nn0uz 10982 . . 3  |-  NN0  =  ( ZZ>= `  0 )
33 0zd 10745 . . 3  |-  ( ph  ->  0  e.  ZZ )
34 0p1e1 10520 . . . . . . 7  |-  ( 0  +  1 )  =  1
3534fveq2i 5778 . . . . . 6  |-  ( ZZ>= `  ( 0  +  1 ) )  =  (
ZZ>= `  1 )
36 nnuz 10983 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3735, 36eqtr4i 2481 . . . . 5  |-  ( ZZ>= `  ( 0  +  1 ) )  =  NN
3837eleq2i 2526 . . . 4  |-  ( z  e.  ( ZZ>= `  (
0  +  1 ) )  <->  z  e.  NN )
399equncomi 3586 . . . . . . . 8  |-  C  =  ( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } )
4039fveq1i 5776 . . . . . . 7  |-  ( C `
 z )  =  ( ( F  u.  {
<. 0 ,  <. 0 ,  1 >. >. } ) `  z
)
41 nnne0 10441 . . . . . . . . 9  |-  ( z  e.  NN  ->  z  =/=  0 )
4241necomd 2716 . . . . . . . 8  |-  ( z  e.  NN  ->  0  =/=  z )
43 fvunsn 5995 . . . . . . . 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 2502 . . . . . 6  |-  ( z  e.  NN  ->  ( C `  z )  =  ( F `  z ) )
4645adantl 466 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  =  ( F `  z
) )
477ffvelrnda 5928 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( F `
 z )  e.  RR )
4846, 47eqeltrd 2536 . . . 4  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  e.  RR )
4938, 48sylan2b 475 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( C `  z )  e.  RR )
5016, 31, 32, 33, 49seqf2 11912 . 2  |-  ( ph  ->  seq 0 ( D ,  C ) : NN0 --> ( RR  X.  RR ) )
511feq1i 5635 . 2  |-  ( G : NN0 --> ( RR 
X.  RR )  <->  seq 0
( D ,  C
) : NN0 --> ( RR 
X.  RR ) )
5250, 51sylibr 212 1  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757    =/= wne 2641   [_csb 3372    u. cun 3410   ifcif 3875   {csn 3961   <.cop 3967   class class class wbr 4376    X. cxp 4922   -->wf 5498   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   1stc1st 6661   2ndc2nd 6662   RRcr 9368   0cc0 9369   1c1 9370    + caddc 9372    < clt 9505    / cdiv 10080   NNcn 10409   2c2 10458   NN0cn0 10666   ZZcz 10733   ZZ>=cuz 10948    seqcseq 11893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-2 10467  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-seq 11894
This theorem is referenced by:  ruclem8  13607  ruclem9  13608  ruclem10  13609  ruclem11  13610  ruclem12  13611
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