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Theorem seqof 11103
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 5539 . . . . . . . . 9  |-  ( G `
 x )  e. 
_V
32rgenw 2610 . . . . . . . 8  |-  A. z  e.  A  ( G `  x )  e.  _V
4 eqid 2283 . . . . . . . . 9  |-  ( z  e.  A  |->  ( G `
 x ) )  =  ( z  e.  A  |->  ( G `  x ) )
54fnmpt 5370 . . . . . . . 8  |-  ( A. z  e.  A  ( G `  x )  e.  _V  ->  ( z  e.  A  |->  ( G `
 x ) )  Fn  A )
63, 5mp1i 11 . . . . . . 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 5335 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( ( F `  x )  Fn  A  <->  ( z  e.  A  |->  ( G `  x ) )  Fn  A ) )
96, 8mpbird 223 . . . . . 6  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  Fn  A
)
10 fvex 5539 . . . . . . 7  |-  ( F `
 x )  e. 
_V
11 fneq1 5333 . . . . . . 7  |-  ( z  =  ( F `  x )  ->  (
z  Fn  A  <->  ( F `  x )  Fn  A
) )
1210, 11elab 2914 . . . . . 6  |-  ( ( F `  x )  e.  { z  |  z  Fn  A }  <->  ( F `  x )  Fn  A )
139, 12sylibr 203 . . . . 5  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  {
z  |  z  Fn  A } )
14 simprl 732 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  ->  x  Fn  A )
15 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  -> 
y  Fn  A )
16 seqof.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  V )
1716adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  ->  A  e.  V )
18 inidm 3378 . . . . . . . . 9  |-  ( A  i^i  A )  =  A
1914, 15, 17, 17, 18offn 6089 . . . . . . . 8  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  -> 
( x  o F 
.+  y )  Fn  A )
2019ex 423 . . . . . . 7  |-  ( ph  ->  ( ( x  Fn  A  /\  y  Fn  A )  ->  (
x  o F  .+  y )  Fn  A
) )
21 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
22 fneq1 5333 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  Fn  A  <->  x  Fn  A ) )
2321, 22elab 2914 . . . . . . . 8  |-  ( x  e.  { z  |  z  Fn  A }  <->  x  Fn  A )
24 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
25 fneq1 5333 . . . . . . . . 9  |-  ( z  =  y  ->  (
z  Fn  A  <->  y  Fn  A ) )
2624, 25elab 2914 . . . . . . . 8  |-  ( y  e.  { z  |  z  Fn  A }  <->  y  Fn  A )
2723, 26anbi12i 678 . . . . . . 7  |-  ( ( x  e.  { z  |  z  Fn  A }  /\  y  e.  {
z  |  z  Fn  A } )  <->  ( x  Fn  A  /\  y  Fn  A ) )
28 ovex 5883 . . . . . . . 8  |-  ( x  o F  .+  y
)  e.  _V
29 fneq1 5333 . . . . . . . 8  |-  ( z  =  ( x  o F  .+  y )  ->  ( z  Fn  A  <->  ( x  o F  .+  y )  Fn  A ) )
3028, 29elab 2914 . . . . . . 7  |-  ( ( x  o F  .+  y )  e.  {
z  |  z  Fn  A }  <->  ( x  o F  .+  y )  Fn  A )
3120, 27, 303imtr4g 261 . . . . . 6  |-  ( ph  ->  ( ( x  e. 
{ z  |  z  Fn  A }  /\  y  e.  { z  |  z  Fn  A } )  ->  (
x  o F  .+  y )  e.  {
z  |  z  Fn  A } ) )
3231imp 418 . . . . 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 11066 . . . 4  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  e.  { z  |  z  Fn  A } )
34 fvex 5539 . . . . 5  |-  (  seq 
M (  o F 
.+  ,  F ) `
 N )  e. 
_V
35 fneq1 5333 . . . . 5  |-  ( z  =  (  seq  M
(  o F  .+  ,  F ) `  N
)  ->  ( z  Fn  A  <->  (  seq  M
(  o F  .+  ,  F ) `  N
)  Fn  A ) )
3634, 35elab 2914 . . . 4  |-  ( (  seq  M (  o F  .+  ,  F
) `  N )  e.  { z  |  z  Fn  A }  <->  (  seq  M (  o F  .+  ,  F ) `  N
)  Fn  A )
3733, 36sylib 188 . . 3  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  Fn  A )
38 dffn5 5568 . . 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 188 . 2  |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  ( (  seq  M (  o F  .+  ,  F
) `  N ) `  z ) ) )
40 fveq1 5524 . . . . . 6  |-  ( w  =  (  seq  M
(  o F  .+  ,  F ) `  N
)  ->  ( w `  z )  =  ( (  seq  M (  o F  .+  ,  F ) `  N
) `  z )
)
41 eqid 2283 . . . . . 6  |-  ( w  e.  _V  |->  ( w `
 z ) )  =  ( w  e. 
_V  |->  ( w `  z ) )
42 fvex 5539 . . . . . 6  |-  ( (  seq  M (  o F  .+  ,  F
) `  N ) `  z )  e.  _V
4340, 41, 42fvmpt 5602 . . . . 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 11 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  (  seq  M (  o F 
.+  ,  F ) `
 N ) )  =  ( (  seq 
M (  o F 
.+  ,  F ) `
 N ) `  z ) )
4532adantlr 695 . . . . 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 695 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  { z  |  z  Fn  A } )
471adantr 451 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  N  e.  ( ZZ>= `  M )
)
48 eqidd 2284 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
x `  z )  =  ( x `  z ) )
49 eqidd 2284 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
y `  z )  =  ( y `  z ) )
5014, 15, 17, 17, 18, 48, 49ofval 6087 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
( x  o F 
.+  y ) `  z )  =  ( ( x `  z
)  .+  ( y `  z ) ) )
5150an32s 779 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  Fn  A  /\  y  Fn  A )
)  ->  ( (
x  o F  .+  y ) `  z
)  =  ( ( x `  z ) 
.+  ( y `  z ) ) )
52 fveq1 5524 . . . . . . . . 9  |-  ( w  =  ( x  o F  .+  y )  ->  ( w `  z )  =  ( ( x  o F 
.+  y ) `  z ) )
53 fvex 5539 . . . . . . . . 9  |-  ( ( x  o F  .+  y ) `  z
)  e.  _V
5452, 41, 53fvmpt 5602 . . . . . . . 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 5524 . . . . . . . . . 10  |-  ( w  =  x  ->  (
w `  z )  =  ( x `  z ) )
57 fvex 5539 . . . . . . . . . 10  |-  ( x `
 z )  e. 
_V
5856, 41, 57fvmpt 5602 . . . . . . . . 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 5524 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w `  z )  =  ( y `  z ) )
61 fvex 5539 . . . . . . . . . 10  |-  ( y `
 z )  e. 
_V
6260, 41, 61fvmpt 5602 . . . . . . . . 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 5870 . . . . . . 7  |-  ( ( ( w  e.  _V  |->  ( w `  z
) ) `  x
)  .+  ( (
w  e.  _V  |->  ( w `  z ) ) `  y ) )  =  ( ( x `  z ) 
.+  ( y `  z ) )
6551, 55, 643eqtr4g 2340 . . . . . 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 461 . . . . 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 5524 . . . . . . . 8  |-  ( w  =  ( F `  x )  ->  (
w `  z )  =  ( ( F `
 x ) `  z ) )
68 fvex 5539 . . . . . . . 8  |-  ( ( F `  x ) `
 z )  e. 
_V
6967, 41, 68fvmpt 5602 . . . . . . 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 695 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  =  ( z  e.  A  |->  ( G `  x ) ) )
7271fveq1d 5527 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( F `  x
) `  z )  =  ( ( z  e.  A  |->  ( G `
 x ) ) `
 z ) )
73 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  z  e.  A )
744fvmpt2 5608 . . . . . . . 8  |-  ( ( z  e.  A  /\  ( G `  x )  e.  _V )  -> 
( ( z  e.  A  |->  ( G `  x ) ) `  z )  =  ( G `  x ) )
7573, 2, 74sylancl 643 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( z  e.  A  |->  ( G `  x
) ) `  z
)  =  ( G `
 x ) )
7672, 75eqtrd 2315 . . . . . 6  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( F `  x
) `  z )  =  ( G `  x ) )
7770, 76syl5eq 2327 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  ( F `  x )
)  =  ( G `
 x ) )
7845, 46, 47, 66, 77seqhomo 11093 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  (  seq  M (  o F 
.+  ,  F ) `
 N ) )  =  (  seq  M
(  .+  ,  G
) `  N )
)
7944, 78eqtr3d 2317 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  (
(  seq  M (  o F  .+  ,  F
) `  N ) `  z )  =  (  seq  M (  .+  ,  G ) `  N
) )
8079mpteq2dva 4106 . 2  |-  ( ph  ->  ( z  e.  A  |->  ( (  seq  M
(  o F  .+  ,  F ) `  N
) `  z )
)  =  ( z  e.  A  |->  (  seq 
M (  .+  ,  G ) `  N
) ) )
8139, 80eqtrd 2315 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 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788    e. cmpt 4077    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    o Fcof 6076   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046
This theorem is referenced by:  mtest  19781  pserulm  19798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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