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Theorem seqfeq2 11301
Description: Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqfveq2.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
seqfveq2.2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
seqfeq2.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
seqfeq2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq  K
(  .+  ,  G
) )
Distinct variable groups:    k, F    k, G    k, K    ph, k
Allowed substitution hints:    .+ ( k)    M( k)

Proof of Theorem seqfeq2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 seqfveq2.1 . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
2 eluzel2 10449 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3 seqfn 11290 . . . 4  |-  ( M  e.  ZZ  ->  seq  M (  .+  ,  F
)  Fn  ( ZZ>= `  M ) )
41, 2, 33syl 19 . . 3  |-  ( ph  ->  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
5 uzss 10462 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  M ) )
61, 5syl 16 . . 3  |-  ( ph  ->  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)
7 fnssres 5517 . . 3  |-  ( (  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M )  /\  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)  ->  (  seq  M (  .+  ,  F
)  |`  ( ZZ>= `  K
) )  Fn  ( ZZ>=
`  K ) )
84, 6, 7syl2anc 643 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  Fn  ( ZZ>= `  K ) )
9 eluzelz 10452 . . 3  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
10 seqfn 11290 . . 3  |-  ( K  e.  ZZ  ->  seq  K (  .+  ,  G
)  Fn  ( ZZ>= `  K ) )
111, 9, 103syl 19 . 2  |-  ( ph  ->  seq  K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
12 fvres 5704 . . . 4  |-  ( x  e.  ( ZZ>= `  K
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )
1312adantl 453 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )
141adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
15 seqfveq2.2 . . . . 5  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
1615adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  ( G `  K ) )
17 simpr 448 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  K )
)
18 elfzuz 11011 . . . . . 6  |-  ( k  e.  ( ( K  +  1 ) ... x )  ->  k  e.  ( ZZ>= `  ( K  +  1 ) ) )
19 seqfeq2.4 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
2018, 19sylan2 461 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2120adantlr 696 . . . 4  |-  ( ( ( ph  /\  x  e.  ( ZZ>= `  K )
)  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2214, 16, 17, 21seqfveq2 11300 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  (  seq  M (  .+  ,  F
) `  x )  =  (  seq  K ( 
.+  ,  G ) `
 x ) )
2313, 22eqtrd 2436 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  K ( 
.+  ,  G ) `
 x ) )
248, 11, 23eqfnfvd 5789 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq  K
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280    |` cres 4839    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   1c1 8947    + caddc 8949   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278
This theorem is referenced by:  seqid  11323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279
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