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Theorem seqof 12167
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 (  oF  .+  ,  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 5882 . . . . . . . . 9  |-  ( G `
 x )  e. 
_V
32rgenw 2818 . . . . . . . 8  |-  A. z  e.  A  ( G `  x )  e.  _V
4 eqid 2457 . . . . . . . . 9  |-  ( z  e.  A  |->  ( G `
 x ) )  =  ( z  e.  A  |->  ( G `  x ) )
54fnmpt 5713 . . . . . . . 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 5677 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( ( F `  x )  Fn  A  <->  ( z  e.  A  |->  ( G `  x ) )  Fn  A ) )
96, 8mpbird 232 . . . . . 6  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  Fn  A
)
10 fvex 5882 . . . . . . 7  |-  ( F `
 x )  e. 
_V
11 fneq1 5675 . . . . . . 7  |-  ( z  =  ( F `  x )  ->  (
z  Fn  A  <->  ( F `  x )  Fn  A
) )
1210, 11elab 3246 . . . . . 6  |-  ( ( F `  x )  e.  { z  |  z  Fn  A }  <->  ( F `  x )  Fn  A )
139, 12sylibr 212 . . . . 5  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  {
z  |  z  Fn  A } )
14 simprl 756 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  ->  x  Fn  A )
15 simprr 757 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  -> 
y  Fn  A )
16 seqof.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  V )
1716adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  ->  A  e.  V )
18 inidm 3703 . . . . . . . . 9  |-  ( A  i^i  A )  =  A
1914, 15, 17, 17, 18offn 6550 . . . . . . . 8  |-  ( (
ph  /\  ( x  Fn  A  /\  y  Fn  A ) )  -> 
( x  oF  .+  y )  Fn  A )
2019ex 434 . . . . . . 7  |-  ( ph  ->  ( ( x  Fn  A  /\  y  Fn  A )  ->  (
x  oF  .+  y )  Fn  A
) )
21 vex 3112 . . . . . . . . 9  |-  x  e. 
_V
22 fneq1 5675 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  Fn  A  <->  x  Fn  A ) )
2321, 22elab 3246 . . . . . . . 8  |-  ( x  e.  { z  |  z  Fn  A }  <->  x  Fn  A )
24 vex 3112 . . . . . . . . 9  |-  y  e. 
_V
25 fneq1 5675 . . . . . . . . 9  |-  ( z  =  y  ->  (
z  Fn  A  <->  y  Fn  A ) )
2624, 25elab 3246 . . . . . . . 8  |-  ( y  e.  { z  |  z  Fn  A }  <->  y  Fn  A )
2723, 26anbi12i 697 . . . . . . 7  |-  ( ( x  e.  { z  |  z  Fn  A }  /\  y  e.  {
z  |  z  Fn  A } )  <->  ( x  Fn  A  /\  y  Fn  A ) )
28 ovex 6324 . . . . . . . 8  |-  ( x  oF  .+  y
)  e.  _V
29 fneq1 5675 . . . . . . . 8  |-  ( z  =  ( x  oF  .+  y )  ->  ( z  Fn  A  <->  ( x  oF  .+  y )  Fn  A ) )
3028, 29elab 3246 . . . . . . 7  |-  ( ( x  oF  .+  y )  e.  {
z  |  z  Fn  A }  <->  ( x  oF  .+  y )  Fn  A )
3120, 27, 303imtr4g 270 . . . . . 6  |-  ( ph  ->  ( ( x  e. 
{ z  |  z  Fn  A }  /\  y  e.  { z  |  z  Fn  A } )  ->  (
x  oF  .+  y )  e.  {
z  |  z  Fn  A } ) )
3231imp 429 . . . . 5  |-  ( (
ph  /\  ( x  e.  { z  |  z  Fn  A }  /\  y  e.  { z  |  z  Fn  A } ) )  -> 
( x  oF  .+  y )  e. 
{ z  |  z  Fn  A } )
331, 13, 32seqcl 12130 . . . 4  |-  ( ph  ->  (  seq M (  oF  .+  ,  F ) `  N
)  e.  { z  |  z  Fn  A } )
34 fvex 5882 . . . . 5  |-  (  seq M (  oF  .+  ,  F ) `
 N )  e. 
_V
35 fneq1 5675 . . . . 5  |-  ( z  =  (  seq M
(  oF  .+  ,  F ) `  N
)  ->  ( z  Fn  A  <->  (  seq M
(  oF  .+  ,  F ) `  N
)  Fn  A ) )
3634, 35elab 3246 . . . 4  |-  ( (  seq M (  oF  .+  ,  F
) `  N )  e.  { z  |  z  Fn  A }  <->  (  seq M (  oF  .+  ,  F ) `
 N )  Fn  A )
3733, 36sylib 196 . . 3  |-  ( ph  ->  (  seq M (  oF  .+  ,  F ) `  N
)  Fn  A )
38 dffn5 5918 . . 3  |-  ( (  seq M (  oF  .+  ,  F
) `  N )  Fn  A  <->  (  seq M
(  oF  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  ( (  seq M (  oF  .+  ,  F
) `  N ) `  z ) ) )
3937, 38sylib 196 . 2  |-  ( ph  ->  (  seq M (  oF  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  ( (  seq M (  oF  .+  ,  F
) `  N ) `  z ) ) )
40 fveq1 5871 . . . . . 6  |-  ( w  =  (  seq M
(  oF  .+  ,  F ) `  N
)  ->  ( w `  z )  =  ( (  seq M (  oF  .+  ,  F ) `  N
) `  z )
)
41 eqid 2457 . . . . . 6  |-  ( w  e.  _V  |->  ( w `
 z ) )  =  ( w  e. 
_V  |->  ( w `  z ) )
42 fvex 5882 . . . . . 6  |-  ( (  seq M (  oF  .+  ,  F
) `  N ) `  z )  e.  _V
4340, 41, 42fvmpt 5956 . . . . 5  |-  ( (  seq M (  oF  .+  ,  F
) `  N )  e.  _V  ->  ( (
w  e.  _V  |->  ( w `  z ) ) `  (  seq M (  oF  .+  ,  F ) `
 N ) )  =  ( (  seq M (  oF  .+  ,  F ) `
 N ) `  z ) )
4434, 43mp1i 12 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  (  seq M (  oF  .+  ,  F ) `
 N ) )  =  ( (  seq M (  oF  .+  ,  F ) `
 N ) `  z ) )
4532adantlr 714 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  e.  { z  |  z  Fn  A }  /\  y  e.  {
z  |  z  Fn  A } ) )  ->  ( x  oF  .+  y )  e.  { z  |  z  Fn  A }
)
4613adantlr 714 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  { z  |  z  Fn  A } )
471adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  N  e.  ( ZZ>= `  M )
)
48 eqidd 2458 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
x `  z )  =  ( x `  z ) )
49 eqidd 2458 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
y `  z )  =  ( y `  z ) )
5014, 15, 17, 17, 18, 48, 49ofval 6548 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  Fn  A  /\  y  Fn  A )
)  /\  z  e.  A )  ->  (
( x  oF  .+  y ) `  z )  =  ( ( x `  z
)  .+  ( y `  z ) ) )
5150an32s 804 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  Fn  A  /\  y  Fn  A )
)  ->  ( (
x  oF  .+  y ) `  z
)  =  ( ( x `  z ) 
.+  ( y `  z ) ) )
52 fveq1 5871 . . . . . . . . 9  |-  ( w  =  ( x  oF  .+  y )  ->  ( w `  z )  =  ( ( x  oF  .+  y ) `  z ) )
53 fvex 5882 . . . . . . . . 9  |-  ( ( x  oF  .+  y ) `  z
)  e.  _V
5452, 41, 53fvmpt 5956 . . . . . . . 8  |-  ( ( x  oF  .+  y )  e.  _V  ->  ( ( w  e. 
_V  |->  ( w `  z ) ) `  ( x  oF  .+  y ) )  =  ( ( x  oF  .+  y ) `
 z ) )
5528, 54ax-mp 5 . . . . . . 7  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  ( x  oF  .+  y
) )  =  ( ( x  oF  .+  y ) `  z )
56 fveq1 5871 . . . . . . . . . 10  |-  ( w  =  x  ->  (
w `  z )  =  ( x `  z ) )
57 fvex 5882 . . . . . . . . . 10  |-  ( x `
 z )  e. 
_V
5856, 41, 57fvmpt 5956 . . . . . . . . 9  |-  ( x  e.  _V  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  x
)  =  ( x `
 z ) )
5921, 58ax-mp 5 . . . . . . . 8  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  x )  =  ( x `  z )
60 fveq1 5871 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w `  z )  =  ( y `  z ) )
61 fvex 5882 . . . . . . . . . 10  |-  ( y `
 z )  e. 
_V
6260, 41, 61fvmpt 5956 . . . . . . . . 9  |-  ( y  e.  _V  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  y
)  =  ( y `
 z ) )
6324, 62ax-mp 5 . . . . . . . 8  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  y )  =  ( y `  z )
6459, 63oveq12i 6308 . . . . . . 7  |-  ( ( ( w  e.  _V  |->  ( w `  z
) ) `  x
)  .+  ( (
w  e.  _V  |->  ( w `  z ) ) `  y ) )  =  ( ( x `  z ) 
.+  ( y `  z ) )
6551, 55, 643eqtr4g 2523 . . . . . 6  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  Fn  A  /\  y  Fn  A )
)  ->  ( (
w  e.  _V  |->  ( w `  z ) ) `  ( x  oF  .+  y
) )  =  ( ( ( w  e. 
_V  |->  ( w `  z ) ) `  x )  .+  (
( w  e.  _V  |->  ( w `  z
) ) `  y
) ) )
6627, 65sylan2b 475 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  (
x  e.  { z  |  z  Fn  A }  /\  y  e.  {
z  |  z  Fn  A } ) )  ->  ( ( w  e.  _V  |->  ( w `
 z ) ) `
 ( x  oF  .+  y ) )  =  ( ( ( w  e.  _V  |->  ( w `  z
) ) `  x
)  .+  ( (
w  e.  _V  |->  ( w `  z ) ) `  y ) ) )
67 fveq1 5871 . . . . . . . 8  |-  ( w  =  ( F `  x )  ->  (
w `  z )  =  ( ( F `
 x ) `  z ) )
68 fvex 5882 . . . . . . . 8  |-  ( ( F `  x ) `
 z )  e. 
_V
6967, 41, 68fvmpt 5956 . . . . . . 7  |-  ( ( F `  x )  e.  _V  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  ( F `  x )
)  =  ( ( F `  x ) `
 z ) )
7010, 69ax-mp 5 . . . . . 6  |-  ( ( w  e.  _V  |->  ( w `  z ) ) `  ( F `
 x ) )  =  ( ( F `
 x ) `  z )
717adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  =  ( z  e.  A  |->  ( G `  x ) ) )
7271fveq1d 5874 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( F `  x
) `  z )  =  ( ( z  e.  A  |->  ( G `
 x ) ) `
 z ) )
73 simplr 755 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  z  e.  A )
744fvmpt2 5964 . . . . . . . 8  |-  ( ( z  e.  A  /\  ( G `  x )  e.  _V )  -> 
( ( z  e.  A  |->  ( G `  x ) ) `  z )  =  ( G `  x ) )
7573, 2, 74sylancl 662 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( z  e.  A  |->  ( G `  x
) ) `  z
)  =  ( G `
 x ) )
7672, 75eqtrd 2498 . . . . . 6  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( F `  x
) `  z )  =  ( G `  x ) )
7770, 76syl5eq 2510 . . . . 5  |-  ( ( ( ph  /\  z  e.  A )  /\  x  e.  ( M ... N
) )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  ( F `  x )
)  =  ( G `
 x ) )
7845, 46, 47, 66, 77seqhomo 12157 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  (
( w  e.  _V  |->  ( w `  z
) ) `  (  seq M (  oF  .+  ,  F ) `
 N ) )  =  (  seq M
(  .+  ,  G
) `  N )
)
7944, 78eqtr3d 2500 . . 3  |-  ( (
ph  /\  z  e.  A )  ->  (
(  seq M (  oF  .+  ,  F
) `  N ) `  z )  =  (  seq M (  .+  ,  G ) `  N
) )
8079mpteq2dva 4543 . 2  |-  ( ph  ->  ( z  e.  A  |->  ( (  seq M
(  oF  .+  ,  F ) `  N
) `  z )
)  =  ( z  e.  A  |->  (  seq M (  .+  ,  G ) `  N
) ) )
8139, 80eqtrd 2498 1  |-  ( ph  ->  (  seq M (  oF  .+  ,  F ) `  N
)  =  ( z  e.  A  |->  (  seq M (  .+  ,  G ) `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   _Vcvv 3109    |-> cmpt 4515    Fn wfn 5589   ` cfv 5594  (class class class)co 6296    oFcof 6537   ZZ>=cuz 11106   ...cfz 11697    seqcseq 12110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111
This theorem is referenced by:  seqof2  12168  mtest  22925  pserulm  22943
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