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Theorem seqfeq2 11158
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 10324 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3 seqfn 11147 . . . 4  |-  ( M  e.  ZZ  ->  seq  M (  .+  ,  F
)  Fn  ( ZZ>= `  M ) )
41, 2, 33syl 18 . . 3  |-  ( ph  ->  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
5 uzss 10337 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  M ) )
61, 5syl 15 . . 3  |-  ( ph  ->  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)
7 fnssres 5436 . . 3  |-  ( (  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M )  /\  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)  ->  (  seq  M (  .+  ,  F
)  |`  ( ZZ>= `  K
) )  Fn  ( ZZ>=
`  K ) )
84, 6, 7syl2anc 642 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  Fn  ( ZZ>= `  K ) )
9 eluzelz 10327 . . 3  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
10 seqfn 11147 . . 3  |-  ( K  e.  ZZ  ->  seq  K (  .+  ,  G
)  Fn  ( ZZ>= `  K ) )
111, 9, 103syl 18 . 2  |-  ( ph  ->  seq  K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
12 fvres 5622 . . . 4  |-  ( x  e.  ( ZZ>= `  K
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )
1312adantl 452 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )
141adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
15 seqfveq2.2 . . . . 5  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
1615adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  ( G `  K ) )
17 simpr 447 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  K )
)
18 elfzuz 10883 . . . . . 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 460 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2120adantlr 695 . . . 4  |-  ( ( ( ph  /\  x  e.  ( ZZ>= `  K )
)  /\  k  e.  ( ( K  + 
1 ) ... x
) )  ->  ( F `  k )  =  ( G `  k ) )
2214, 16, 17, 21seqfveq2 11157 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  (  seq  M (  .+  ,  F
) `  x )  =  (  seq  K ( 
.+  ,  G ) `
 x ) )
2313, 22eqtrd 2390 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( (  seq  M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 x )  =  (  seq  K ( 
.+  ,  G ) `
 x ) )
248, 11, 23eqfnfvd 5705 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq  K
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228    |` cres 4770    Fn wfn 5329   ` cfv 5334  (class class class)co 5942   1c1 8825    + caddc 8827   ZZcz 10113   ZZ>=cuz 10319   ...cfz 10871    seq cseq 11135
This theorem is referenced by:  seqid  11180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-seq 11136
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