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Theorem seqfeq3 10253
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 2117 . . . 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 10202 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
65ffnd 5243 . 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 2195 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
91, 2, 3, 8seqf 10202 . . 3  |-  ( ph  ->  seq M ( Q ,  F ) : ( ZZ>= `  M ) --> S )
109ffnd 5243 . 2  |-  ( ph  ->  seq M ( Q ,  F )  Fn  ( ZZ>= `  M )
)
115ffvelrnda 5523 . . . 4  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  a
)  e.  S )
12 fvi 5446 . . . 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 468 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
153adantlr 468 . . . 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 468 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
18 fvi 5446 . . . . . 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 5446 . . . . . . 7  |-  ( x  e.  S  ->  (  _I  `  x )  =  x )
2120ad2antrl 481 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  x
)  =  x )
22 fvi 5446 . . . . . . 7  |-  ( y  e.  S  ->  (  _I  `  y )  =  y )
2322ad2antll 482 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  y
)  =  y )
2421, 23oveq12d 5760 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( (  _I  `  x ) Q (  _I  `  y ) )  =  ( x Q y ) )
2517, 19, 243eqtr4d 2160 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  _I  `  (
x  .+  y )
)  =  ( (  _I  `  x ) Q (  _I  `  y ) ) )
26 fvi 5446 . . . . 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 468 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
2914, 15, 16, 25, 27, 15, 28seq3homo 10251 . . 3  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  _I  `  (  seq M ( 
.+  ,  F ) `
 a ) )  =  (  seq M
( Q ,  F
) `  a )
)
3013, 29eqtr3d 2152 . 2  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  a
)  =  (  seq M ( Q ,  F ) `  a
) )
316, 10, 30eqfnfvd 5489 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M ( Q ,  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465    _I cid 4180   ` cfv 5093  (class class class)co 5742   ZZcz 9022   ZZ>=cuz 9294    seqcseq 10186
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-seqfrec 10187
This theorem is referenced by: (None)
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