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Theorem seqfeq3 10461
Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
seqfeq3.m  |-  ( ph  ->  M  e.  ZZ )
seqfeq3.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seqfeq3.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seqfeq3.id  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
Assertion
Ref Expression
seqfeq3  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M ( Q ,  F ) )
Distinct variable groups:    ph, x, y   
x, F, y    x, M, y    x,  .+ , y    x, Q, y    x, S, y

Proof of Theorem seqfeq3
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqid 2170 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 seqfeq3.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 seqfeq3.f . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
4 seqfeq3.cl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
51, 2, 3, 4seqf 10410 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
65ffnd 5346 . 2  |-  ( ph  ->  seq M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
7 seqfeq3.id . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
87, 4eqeltrrd 2248 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
91, 2, 3, 8seqf 10410 . . 3  |-  ( ph  ->  seq M ( Q ,  F ) : ( ZZ>= `  M ) --> S )
109ffnd 5346 . 2  |-  ( ph  ->  seq M ( Q ,  F )  Fn  ( ZZ>= `  M )
)
115ffvelrnda 5629 . . . 4  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  a
)  e.  S )
12 fvi 5551 . . . 4  |-  ( (  seq M (  .+  ,  F ) `  a
)  e.  S  -> 
(  _I  `  (  seq M (  .+  ,  F ) `  a
) )  =  (  seq M (  .+  ,  F ) `  a
) )
1311, 12syl 14 . . 3  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  _I  `  (  seq M ( 
.+  ,  F ) `
 a ) )  =  (  seq M
(  .+  ,  F
) `  a )
)
144adantlr 474 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
153adantlr 474 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
16 simpr 109 . . . 4  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  a  e.  ( ZZ>= `  M )
)
177adantlr 474 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
18 fvi 5551 . . . . . 6  |-  ( ( x  .+  y )  e.  S  ->  (  _I  `  ( x  .+  y ) )  =  ( x  .+  y
) )
1914, 18syl 14 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  (
x  .+  y )
)  =  ( x 
.+  y ) )
20 fvi 5551 . . . . . . 7  |-  ( x  e.  S  ->  (  _I  `  x )  =  x )
2120ad2antrl 487 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  x
)  =  x )
22 fvi 5551 . . . . . . 7  |-  ( y  e.  S  ->  (  _I  `  y )  =  y )
2322ad2antll 488 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  y
)  =  y )
2421, 23oveq12d 5869 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( (  _I  `  x ) Q (  _I  `  y ) )  =  ( x Q y ) )
2517, 19, 243eqtr4d 2213 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  (
x  .+  y )
)  =  ( (  _I  `  x ) Q (  _I  `  y ) ) )
26 fvi 5551 . . . . 5  |-  ( ( F `  x )  e.  S  ->  (  _I  `  ( F `  x ) )  =  ( F `  x
) )
2715, 26syl 14 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  (  _I  `  ( F `  x
) )  =  ( F `  x ) )
288adantlr 474 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
2914, 15, 16, 25, 27, 15, 28seq3homo 10459 . . 3  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  _I  `  (  seq M ( 
.+  ,  F ) `
 a ) )  =  (  seq M
( Q ,  F
) `  a )
)
3013, 29eqtr3d 2205 . 2  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  a
)  =  (  seq M ( Q ,  F ) `  a
) )
316, 10, 30eqfnfvd 5594 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M ( Q ,  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141    _I cid 4271   ` cfv 5196  (class class class)co 5851   ZZcz 9205   ZZ>=cuz 9480    seqcseq 10394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-ltadd 7883
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-inn 8872  df-n0 9129  df-z 9206  df-uz 9481  df-seqfrec 10395
This theorem is referenced by: (None)
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