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Theorem ruclem7 13622
Description: Lemma for ruc 13629. 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 461 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 )  ->  N  e.  NN0 )
2 nn0uz 10998 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2549 . . . 4  |-  ( (
ph  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= `  0 )
)
4 seqp1 11924 . . . 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 5792 . . 3  |-  ( G `
 ( N  + 
1 ) )  =  (  seq 0 ( D ,  C ) `
 ( N  + 
1 ) )
86fveq1i 5792 . . . 4  |-  ( G `
 N )  =  (  seq 0 ( D ,  C ) `
 N )
98oveq1i 6202 . . 3  |-  ( ( G `  N ) D ( C `  ( N  +  1
) ) )  =  ( (  seq 0
( D ,  C
) `  N ) D ( C `  ( N  +  1
) ) )
105, 7, 93eqtr4g 2517 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( G `  ( N  +  1 ) )  =  ( ( G `  N
) D ( C `
 ( N  + 
1 ) ) ) )
11 nn0p1nn 10722 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
1211adantl 466 . . . . . 6  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( N  +  1 )  e.  NN )
1312nnne0d 10469 . . . . 5  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( N  +  1 )  =/=  0 )
1413necomd 2719 . . . 4  |-  ( (
ph  /\  N  e.  NN0 )  ->  0  =/=  ( N  +  1
) )
15 ruc.4 . . . . . . 7  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
1615equncomi 3602 . . . . . 6  |-  C  =  ( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } )
1716fveq1i 5792 . . . . 5  |-  ( C `
 ( N  + 
1 ) )  =  ( ( F  u.  {
<. 0 ,  <. 0 ,  1 >. >. } ) `  ( N  +  1 ) )
18 fvunsn 6011 . . . . 5  |-  ( 0  =/=  ( N  + 
1 )  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 ( N  + 
1 ) )  =  ( F `  ( N  +  1 ) ) )
1917, 18syl5eq 2504 . . . 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 6208 . 2  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( ( G `  N ) D ( C `  ( N  +  1
) ) )  =  ( ( G `  N ) D ( F `  ( N  +  1 ) ) ) )
2210, 21eqtrd 2492 1  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( G `  ( N  +  1 ) )  =  ( ( G `  N
) D ( F `
 ( N  + 
1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   [_csb 3388    u. cun 3426   ifcif 3891   {csn 3977   <.cop 3983   class class class wbr 4392    X. cxp 4938   -->wf 5514   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   1stc1st 6677   2ndc2nd 6678   RRcr 9384   0cc0 9385   1c1 9386    + caddc 9388    < clt 9521    / cdiv 10096   NNcn 10425   2c2 10474   NN0cn0 10682   ZZ>=cuz 10964    seqcseq 11909
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-seq 11910
This theorem is referenced by:  ruclem8  13623  ruclem9  13624  ruclem12  13627
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