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Theorem ruclem7 12835
Description: Lemma for ruc 12842. Successor value for 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
ruclem7  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( G `  ( N  +  1 ) )  =  ( ( G `  N
) D ( F `
 ( N  + 
1 ) ) ) )
Distinct variable groups:    x, m, y, F    m, G, x, y    m, N, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem7
StepHypRef Expression
1 simpr 448 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 nn0uz 10520 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2526 . . . 4  |-  ( (
ph  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
4 seqp1 11338 . . . 4  |-  ( N  e.  ( ZZ>= `  0
)  ->  (  seq  0 ( D ,  C ) `  ( N  +  1 ) )  =  ( (  seq  0 ( D ,  C ) `  N ) D ( C `  ( N  +  1 ) ) ) )
53, 4syl 16 . . 3  |-  ( (
ph  /\  N  e.  NN0 )  ->  (  seq  0 ( D ,  C ) `  ( N  +  1 ) )  =  ( (  seq  0 ( D ,  C ) `  N ) D ( C `  ( N  +  1 ) ) ) )
6 ruc.5 . . . 4  |-  G  =  seq  0 ( D ,  C )
76fveq1i 5729 . . 3  |-  ( G `
 ( N  + 
1 ) )  =  (  seq  0 ( D ,  C ) `
 ( N  + 
1 ) )
86fveq1i 5729 . . . 4  |-  ( G `
 N )  =  (  seq  0 ( D ,  C ) `
 N )
98oveq1i 6091 . . 3  |-  ( ( G `  N ) D ( C `  ( N  +  1
) ) )  =  ( (  seq  0
( D ,  C
) `  N ) D ( C `  ( N  +  1
) ) )
105, 7, 93eqtr4g 2493 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( G `  ( N  +  1 ) )  =  ( ( G `  N
) D ( C `
 ( N  + 
1 ) ) ) )
11 nn0p1nn 10259 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
1211adantl 453 . . . . . 6  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( N  +  1 )  e.  NN )
1312nnne0d 10044 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( N  +  1 )  =/=  0 )
1413necomd 2687 . . . 4  |-  ( (
ph  /\  N  e.  NN0 )  ->  0  =/=  ( N  +  1
) )
15 ruc.4 . . . . . . 7  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
1615equncomi 3493 . . . . . 6  |-  C  =  ( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } )
1716fveq1i 5729 . . . . 5  |-  ( C `
 ( N  + 
1 ) )  =  ( ( F  u.  {
<. 0 ,  <. 0 ,  1 >. >. } ) `  ( N  +  1 ) )
18 fvunsn 5925 . . . . 5  |-  ( 0  =/=  ( N  + 
1 )  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 ( N  + 
1 ) )  =  ( F `  ( N  +  1 ) ) )
1917, 18syl5eq 2480 . . . 4  |-  ( 0  =/=  ( N  + 
1 )  ->  ( C `  ( N  +  1 ) )  =  ( F `  ( N  +  1
) ) )
2014, 19syl 16 . . 3  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( C `  ( N  +  1 ) )  =  ( F `  ( N  +  1 ) ) )
2120oveq2d 6097 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( ( G `  N ) D ( C `  ( N  +  1
) ) )  =  ( ( G `  N ) D ( F `  ( N  +  1 ) ) ) )
2210, 21eqtrd 2468 1  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( G `  ( N  +  1 ) )  =  ( ( G `  N
) D ( F `
 ( N  + 
1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   [_csb 3251    u. cun 3318   ifcif 3739   {csn 3814   <.cop 3817   class class class wbr 4212    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZ>=cuz 10488    seq cseq 11323
This theorem is referenced by:  ruclem8  12836  ruclem9  12837  ruclem12  12840
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324
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