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Theorem seqof 11380
Description: Distribute function operation through a sequence. Note that  G ( z ) is an implicit function on  z. (Contributed by Mario Carneiro, 3-Mar-2015.)
Hypotheses
Ref Expression
seqof.1  |-  ( ph  ->  A  e.  V )
seqof.2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
seqof.3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  =  ( z  e.  A  |->  ( G `  x ) ) )
Assertion
Ref Expression
seqof  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  (  seq 
M (  .+  ,  G ) `  N
) ) )
Distinct variable groups:    x, z, A    x, F, z    x, G    x, M, z    x, N, z    x,  .+ , z    ph, x, z
Allowed substitution hints:    G( z)    V( x, z)

Proof of Theorem seqof
Dummy variables  y  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqof.2 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 fvex 5742 . . . . . . . . 9  |-  ( G `
 x )  e. 
_V
32rgenw 2773 . . . . . . . 8  |-  A. z  e.  A  ( G `  x )  e.  _V
4 eqid 2436 . . . . . . . . 9  |-  ( z  e.  A  |->  ( G `
 x ) )  =  ( z  e.  A  |->  ( G `  x ) )
54fnmpt 5571 . . . . . . . 8  |-  ( A. z  e.  A  ( G `  x )  e.  _V  ->  ( z  e.  A  |->  ( G `
 x ) )  Fn  A )
63, 5mp1i 12 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( z  e.  A  |->  ( G `
 x ) )  Fn  A )
7 seqof.3 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  =  ( z  e.  A  |->  ( G `  x ) ) )
87fneq1d 5536 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( ( F `  x )  Fn  A  <->  ( z  e.  A  |->  ( G `  x ) )  Fn  A ) )
96, 8mpbird 224 . . . . . 6  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  Fn  A
)
10 fvex 5742 . . . . . . 7  |-  ( F `
 x )  e. 
_V
11 fneq1 5534 . . . . . . 7  |-  ( z  =  ( F `  x )  ->  (
z  Fn  A  <->  ( F `  x )  Fn  A
) )
1210, 11elab 3082 . . . . . 6  |-  ( ( F `  x )  e.  { z  |  z  Fn  A }  <->  ( F `  x )  Fn  A )
139, 12sylibr 204 . . . . 5  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  {
z  |  z  Fn  A } )
14 simprl 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  ->  x  Fn  A )
15 simprr 734 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  -> 
y  Fn  A )
16 seqof.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  V )
1716adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  ->  A  e.  V )
18 inidm 3550 . . . . . . . . 9  |-  ( A  i^i  A )  =  A
1914, 15, 17, 17, 18offn 6316 . . . . . . . 8  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  -> 
( x  o F 
.+  y )  Fn  A )
2019ex 424 . . . . . . 7  |-  ( ph  ->  ( ( x  Fn  A  /\  y  Fn  A )  ->  (
x  o F  .+  y )  Fn  A
) )
21 vex 2959 . . . . . . . . 9  |-  x  e. 
_V
22 fneq1 5534 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  Fn  A  <->  x  Fn  A ) )
2321, 22elab 3082 . . . . . . . 8  |-  ( x  e.  { z  |  z  Fn  A }  <->  x  Fn  A )
24 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
25 fneq1 5534 . . . . . . . . 9  |-  ( z  =  y  ->  (
z  Fn  A  <->  y  Fn  A ) )
2624, 25elab 3082 . . . . . . . 8  |-  ( y  e.  { z  |  z  Fn  A }  <->  y  Fn  A )
2723, 26anbi12i 679 . . . . . . 7  |-  ( ( x  e.  { z  |  z  Fn  A }  /\  y  e.  {
z  |  z  Fn  A } )  <->  ( x  Fn  A  /\  y  Fn  A ) )
28 ovex 6106 . . . . . . . 8  |-  ( x  o F  .+  y
)  e.  _V
29 fneq1 5534 . . . . . . . 8  |-  ( z  =  ( x  o F  .+  y )  ->  ( z  Fn  A  <->  ( x  o F  .+  y )  Fn  A ) )
3028, 29elab 3082 . . . . . . 7  |-  ( ( x  o F  .+  y )  e.  {
z  |  z  Fn  A }  <->  ( x  o F  .+  y )  Fn  A )
3120, 27, 303imtr4g 262 . . . . . 6  |-  ( ph  ->  ( ( x  e. 
{ z  |  z  Fn  A }  /\  y  e.  { z  |  z  Fn  A } )  ->  (
x  o F  .+  y )  e.  {
z  |  z  Fn  A } ) )
3231imp 419 . . . . 5  |-  ( (
ph  /\  ( x  e.  { z  |  z  Fn  A }  /\  y  e.  { z  |  z  Fn  A } ) )  -> 
( x  o F 
.+  y )  e. 
{ z  |  z  Fn  A } )
331, 13, 32seqcl 11343 . . . 4  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  e.  { z  |  z  Fn  A } )
34 fvex 5742 . . . . 5  |-  (  seq 
M (  o F 
.+  ,  F ) `
 N )  e. 
_V
35 fneq1 5534 . . . . 5  |-  ( z  =  (  seq  M
(  o F  .+  ,  F ) `  N
)  ->  ( z  Fn  A  <->  (  seq  M
(  o F  .+  ,  F ) `  N
)  Fn  A ) )
3634, 35elab 3082 . . . 4  |-  ( (  seq  M (  o F  .+  ,  F
) `  N )  e.  { z  |  z  Fn  A }  <->  (  seq  M (  o F  .+  ,  F ) `  N
)  Fn  A )
3733, 36sylib 189 . . 3  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  Fn  A )
38 dffn5 5772 . . 3  |-  ( (  seq  M (  o F  .+  ,  F
) `  N )  Fn  A  <->  (  seq  M
(  o F  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  ( (  seq  M (  o F  .+  ,  F
) `  N ) `  z ) ) )
3937, 38sylib 189 . 2  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  ( (  seq  M (  o F  .+  ,  F
) `  N ) `  z ) ) )
40 fveq1 5727 . . . . . 6  |-  ( w  =  (  seq  M
(  o F  .+  ,  F ) `  N
)  ->  ( w `  z )  =  ( (  seq  M (  o F  .+  ,  F ) `  N
) `  z )
)
41 eqid 2436 . . . . . 6  |-  ( w  e.  _V  |->  ( w `
 z ) )  =  ( w  e. 
_V  |->  ( w `  z ) )
42 fvex 5742 . . . . . 6  |-  ( (  seq  M (  o F  .+  ,  F
) `  N ) `  z )  e.  _V
4340, 41, 42fvmpt 5806 . . . . 5  |-  ( (  seq  M (  o F  .+  ,  F
) `  N )  e.  _V  ->  ( (
w  e.  _V  |->  ( w `  z ) ) `  (  seq 
M (  o F 
.+  ,  F ) `
 N ) )  =  ( (  seq 
M (  o F 
.+  ,  F ) `
 N ) `  z ) )
4434, 43mp1i 12 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  (  seq  M (  o F 
.+  ,  F ) `
 N ) )  =  ( (  seq 
M (  o F 
.+  ,  F ) `
 N ) `  z ) )
4532adantlr 696 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  e.  { z  |  z  Fn  A }  /\  y  e.  {
z  |  z  Fn  A } ) )  ->  ( x  o F  .+  y )  e.  { z  |  z  Fn  A }
)
4613adantlr 696 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  { z  |  z  Fn  A } )
471adantr 452 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  N  e.  ( ZZ>= `  M )
)
48 eqidd 2437 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
x `  z )  =  ( x `  z ) )
49 eqidd 2437 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
y `  z )  =  ( y `  z ) )
5014, 15, 17, 17, 18, 48, 49ofval 6314 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
( x  o F 
.+  y ) `  z )  =  ( ( x `  z
)  .+  ( y `  z ) ) )
5150an32s 780 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  Fn  A  /\  y  Fn  A )
)  ->  ( (
x  o F  .+  y ) `  z
)  =  ( ( x `  z ) 
.+  ( y `  z ) ) )
52 fveq1 5727 . . . . . . . . 9  |-  ( w  =  ( x  o F  .+  y )  ->  ( w `  z )  =  ( ( x  o F 
.+  y ) `  z ) )
53 fvex 5742 . . . . . . . . 9  |-  ( ( x  o F  .+  y ) `  z
)  e.  _V
5452, 41, 53fvmpt 5806 . . . . . . . 8  |-  ( ( x  o F  .+  y )  e.  _V  ->  ( ( w  e. 
_V  |->  ( w `  z ) ) `  ( x  o F  .+  y ) )  =  ( ( x  o F  .+  y ) `
 z ) )
5528, 54ax-mp 8 . . . . . . 7  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  ( x  o F  .+  y
) )  =  ( ( x  o F 
.+  y ) `  z )
56 fveq1 5727 . . . . . . . . . 10  |-  ( w  =  x  ->  (
w `  z )  =  ( x `  z ) )
57 fvex 5742 . . . . . . . . . 10  |-  ( x `
 z )  e. 
_V
5856, 41, 57fvmpt 5806 . . . . . . . . 9  |-  ( x  e.  _V  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  x
)  =  ( x `
 z ) )
5921, 58ax-mp 8 . . . . . . . 8  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  x )  =  ( x `  z )
60 fveq1 5727 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w `  z )  =  ( y `  z ) )
61 fvex 5742 . . . . . . . . . 10  |-  ( y `
 z )  e. 
_V
6260, 41, 61fvmpt 5806 . . . . . . . . 9  |-  ( y  e.  _V  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  y
)  =  ( y `
 z ) )
6324, 62ax-mp 8 . . . . . . . 8  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  y )  =  ( y `  z )
6459, 63oveq12i 6093 . . . . . . 7  |-  ( ( ( w  e.  _V  |->  ( w `  z
) ) `  x
)  .+  ( (
w  e.  _V  |->  ( w `  z ) ) `  y ) )  =  ( ( x `  z ) 
.+  ( y `  z ) )
6551, 55, 643eqtr4g 2493 . . . . . 6  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  Fn  A  /\  y  Fn  A )
)  ->  ( (
w  e.  _V  |->  ( w `  z ) ) `  ( x  o F  .+  y
) )  =  ( ( ( w  e. 
_V  |->  ( w `  z ) ) `  x )  .+  (
( w  e.  _V  |->  ( w `  z
) ) `  y
) ) )
6627, 65sylan2b 462 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  e.  { z  |  z  Fn  A }  /\  y  e.  {
z  |  z  Fn  A } ) )  ->  ( ( w  e.  _V  |->  ( w `
 z ) ) `
 ( x  o F  .+  y ) )  =  ( ( ( w  e.  _V  |->  ( w `  z
) ) `  x
)  .+  ( (
w  e.  _V  |->  ( w `  z ) ) `  y ) ) )
67 fveq1 5727 . . . . . . . 8  |-  ( w  =  ( F `  x )  ->  (
w `  z )  =  ( ( F `
 x ) `  z ) )
68 fvex 5742 . . . . . . . 8  |-  ( ( F `  x ) `
 z )  e. 
_V
6967, 41, 68fvmpt 5806 . . . . . . 7  |-  ( ( F `  x )  e.  _V  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  ( F `  x )
)  =  ( ( F `  x ) `
 z ) )
7010, 69ax-mp 8 . . . . . 6  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  ( F `
 x ) )  =  ( ( F `
 x ) `  z )
717adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  =  ( z  e.  A  |->  ( G `  x ) ) )
7271fveq1d 5730 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( F `  x
) `  z )  =  ( ( z  e.  A  |->  ( G `
 x ) ) `
 z ) )
73 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  z  e.  A )
744fvmpt2 5812 . . . . . . . 8  |-  ( ( z  e.  A  /\  ( G `  x )  e.  _V )  -> 
( ( z  e.  A  |->  ( G `  x ) ) `  z )  =  ( G `  x ) )
7573, 2, 74sylancl 644 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( z  e.  A  |->  ( G `  x
) ) `  z
)  =  ( G `
 x ) )
7672, 75eqtrd 2468 . . . . . 6  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( F `  x
) `  z )  =  ( G `  x ) )
7770, 76syl5eq 2480 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  ( F `  x )
)  =  ( G `
 x ) )
7845, 46, 47, 66, 77seqhomo 11370 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  (  seq  M (  o F 
.+  ,  F ) `
 N ) )  =  (  seq  M
(  .+  ,  G
) `  N )
)
7944, 78eqtr3d 2470 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  (
(  seq  M (  o F  .+  ,  F
) `  N ) `  z )  =  (  seq  M (  .+  ,  G ) `  N
) )
8079mpteq2dva 4295 . 2  |-  ( ph  ->  ( z  e.  A  |->  ( (  seq  M
(  o F  .+  ,  F ) `  N
) `  z )
)  =  ( z  e.  A  |->  (  seq 
M (  .+  ,  G ) `  N
) ) )
8139, 80eqtrd 2468 1  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  (  seq 
M (  .+  ,  G ) `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   _Vcvv 2956    e. cmpt 4266    Fn wfn 5449   ` cfv 5454  (class class class)co 6081    o Fcof 6303   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323
This theorem is referenced by:  seqof2  11381  mtest  20320  pserulm  20338
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-rep 4320  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-of 6305  df-1st 6349  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-fz 11044  df-seq 11324
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