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Theorem seq3feq2 10469
Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
Hypotheses
Ref Expression
seq3fveq2.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
seq3fveq2.2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
seq3fveq2.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seq3fveq2.g  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
seq3fveq2.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seq3feq2.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
seq3feq2  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq K
(  .+  ,  G
) )
Distinct variable groups:    x, k, y, F    k, G, x, y    k, K, x, y    ph, k, x, y   
k, M, x, y    .+ , k, x, y    S, k, x, y

Proof of Theorem seq3feq2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 seq3fveq2.1 . . . . . 6  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
3 eluzel2 9532 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
42, 3syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
5 seq3fveq2.f . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
6 seq3fveq2.pl . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
71, 4, 5, 6seqf 10460 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
87ffnd 5366 . . 3  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
9 uzss 9547 . . . 4  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  M ) )
102, 9syl 14 . . 3  |-  ( ph  ->  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)
11 fnssres 5329 . . 3  |-  ( (  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M )  /\  ( ZZ>= `  K )  C_  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) )  Fn  ( ZZ>= `  K
) )
128, 10, 11syl2anc 411 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  Fn  ( ZZ>= `  K ) )
13 eqid 2177 . . . 4  |-  ( ZZ>= `  K )  =  (
ZZ>= `  K )
14 eluzelz 9536 . . . . 5  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
152, 14syl 14 . . . 4  |-  ( ph  ->  K  e.  ZZ )
16 seq3fveq2.g . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
1713, 15, 16, 6seqf 10460 . . 3  |-  ( ph  ->  seq K (  .+  ,  G ) : (
ZZ>= `  K ) --> S )
1817ffnd 5366 . 2  |-  ( ph  ->  seq K (  .+  ,  G )  Fn  ( ZZ>=
`  K ) )
19 fvres 5539 . . . 4  |-  ( z  e.  ( ZZ>= `  K
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq M ( 
.+  ,  F ) `
 z ) )
2019adantl 277 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq M ( 
.+  ,  F ) `
 z ) )
212adantr 276 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
22 seq3fveq2.2 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
2322adantr 276 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( G `
 K ) )
245adantlr 477 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2516adantlr 477 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
266adantlr 477 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
27 simpr 110 . . . 4  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  z  e.  ( ZZ>= `  K )
)
28 elfzuz 10020 . . . . . 6  |-  ( k  e.  ( ( K  +  1 ) ... z )  ->  k  e.  ( ZZ>= `  ( K  +  1 ) ) )
29 seq3feq2.4 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )
3028, 29sylan2 286 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... z
) )  ->  ( F `  k )  =  ( G `  k ) )
3130adantlr 477 . . . 4  |-  ( ( ( ph  /\  z  e.  ( ZZ>= `  K )
)  /\  k  e.  ( ( K  + 
1 ) ... z
) )  ->  ( F `  k )  =  ( G `  k ) )
3221, 23, 24, 25, 26, 27, 31seq3fveq2 10468 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
) )
3320, 32eqtrd 2210 . 2  |-  ( (
ph  /\  z  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F )  |`  ( ZZ>=
`  K ) ) `
 z )  =  (  seq K ( 
.+  ,  G ) `
 z ) )
3412, 18, 33eqfnfvd 5616 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F )  |`  ( ZZ>= `  K )
)  =  seq K
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    C_ wss 3129    |` cres 4628    Fn wfn 5211   ` cfv 5216  (class class class)co 5874   1c1 7811    + caddc 7813   ZZcz 9252   ZZ>=cuz 9527   ...cfz 10007    seqcseq 10444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-fz 10008  df-seqfrec 10445
This theorem is referenced by:  seq3id  10507
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