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Theorem seqfeq4 11095
Description: Equality of series under different addition operations which agree on an an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
seqfeq4.m  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
seqfeq4.f  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  S
)
seqfeq4.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seqfeq4.id  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
Assertion
Ref Expression
seqfeq4  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  M ( Q ,  F ) `
 N ) )
Distinct variable groups:    x, y,  .+    x, F, y    x, M, y    x, N, y    ph, x, y    x, Q, y    x, S, y

Proof of Theorem seqfeq4
StepHypRef Expression
1 fvex 5539 . . 3  |-  (  seq 
M (  .+  ,  F ) `  N
)  e.  _V
2 fvi 5579 . . 3  |-  ( (  seq  M (  .+  ,  F ) `  N
)  e.  _V  ->  (  _I  `  (  seq 
M (  .+  ,  F ) `  N
) )  =  (  seq  M (  .+  ,  F ) `  N
) )
31, 2ax-mp 8 . 2  |-  (  _I 
`  (  seq  M
(  .+  ,  F
) `  N )
)  =  (  seq 
M (  .+  ,  F ) `  N
)
4 seqfeq4.cl . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
5 seqfeq4.f . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  S
)
6 seqfeq4.m . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
7 seqfeq4.id . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
8 ovex 5883 . . . . 5  |-  ( x 
.+  y )  e. 
_V
9 fvi 5579 . . . . 5  |-  ( ( x  .+  y )  e.  _V  ->  (  _I  `  ( x  .+  y ) )  =  ( x  .+  y
) )
108, 9ax-mp 8 . . . 4  |-  (  _I 
`  ( x  .+  y ) )  =  ( x  .+  y
)
11 vex 2791 . . . . . 6  |-  x  e. 
_V
12 fvi 5579 . . . . . 6  |-  ( x  e.  _V  ->  (  _I  `  x )  =  x )
1311, 12ax-mp 8 . . . . 5  |-  (  _I 
`  x )  =  x
14 vex 2791 . . . . . 6  |-  y  e. 
_V
15 fvi 5579 . . . . . 6  |-  ( y  e.  _V  ->  (  _I  `  y )  =  y )
1614, 15ax-mp 8 . . . . 5  |-  (  _I 
`  y )  =  y
1713, 16oveq12i 5870 . . . 4  |-  ( (  _I  `  x ) Q (  _I  `  y ) )  =  ( x Q y )
187, 10, 173eqtr4g 2340 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  (
x  .+  y )
)  =  ( (  _I  `  x ) Q (  _I  `  y ) ) )
19 fvex 5539 . . . 4  |-  ( F `
 x )  e. 
_V
20 fvi 5579 . . . 4  |-  ( ( F `  x )  e.  _V  ->  (  _I  `  ( F `  x ) )  =  ( F `  x
) )
2119, 20mp1i 11 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  (  _I  `  ( F `  x
) )  =  ( F `  x ) )
224, 5, 6, 18, 21seqhomo 11093 . 2  |-  ( ph  ->  (  _I  `  (  seq  M (  .+  ,  F ) `  N
) )  =  (  seq  M ( Q ,  F ) `  N ) )
233, 22syl5eqr 2329 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  M ( Q ,  F ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    _I cid 4304   ` cfv 5255  (class class class)co 5858   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046
This theorem is referenced by:  seqfeq3  11096  gsumzoppg  15216  gsumpropd2lem  23379
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-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-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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