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Theorem seqfeq4 11374
Description: Equality of series under different addition operations which agree on 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 5744 . . 3  |-  (  seq 
M (  .+  ,  F ) `  N
)  e.  _V
2 fvi 5785 . . 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 6108 . . . . 5  |-  ( x 
.+  y )  e. 
_V
9 fvi 5785 . . . . 5  |-  ( ( x  .+  y )  e.  _V  ->  (  _I  `  ( x  .+  y ) )  =  ( x  .+  y
) )
108, 9ax-mp 8 . . . 4  |-  (  _I 
`  ( x  .+  y ) )  =  ( x  .+  y
)
11 vex 2961 . . . . . 6  |-  x  e. 
_V
12 fvi 5785 . . . . . 6  |-  ( x  e.  _V  ->  (  _I  `  x )  =  x )
1311, 12ax-mp 8 . . . . 5  |-  (  _I 
`  x )  =  x
14 vex 2961 . . . . . 6  |-  y  e. 
_V
15 fvi 5785 . . . . . 6  |-  ( y  e.  _V  ->  (  _I  `  y )  =  y )
1614, 15ax-mp 8 . . . . 5  |-  (  _I 
`  y )  =  y
1713, 16oveq12i 6095 . . . 4  |-  ( (  _I  `  x ) Q (  _I  `  y ) )  =  ( x Q y )
187, 10, 173eqtr4g 2495 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  (
x  .+  y )
)  =  ( (  _I  `  x ) Q (  _I  `  y ) ) )
19 fvex 5744 . . . 4  |-  ( F `
 x )  e. 
_V
20 fvi 5785 . . . 4  |-  ( ( F `  x )  e.  _V  ->  (  _I  `  ( F `  x ) )  =  ( F `  x
) )
2119, 20mp1i 12 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  (  _I  `  ( F `  x
) )  =  ( F `  x ) )
224, 5, 6, 18, 21seqhomo 11372 . 2  |-  ( ph  ->  (  _I  `  (  seq  M (  .+  ,  F ) `  N
) )  =  (  seq  M ( Q ,  F ) `  N ) )
233, 22syl5eqr 2484 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  M ( Q ,  F ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    _I cid 4495   ` cfv 5456  (class class class)co 6083   ZZ>=cuz 10490   ...cfz 11045    seq cseq 11325
This theorem is referenced by:  seqfeq3  11375  gsumzoppg  15541  gsumpropd2lem  24222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-seq 11326
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