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Theorem seqfeq3 11302
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 seqfeq3.m . . 3  |-  ( ph  ->  M  e.  ZZ )
2 seqfn 11264 . . 3  |-  ( M  e.  ZZ  ->  seq  M (  .+  ,  F
)  Fn  ( ZZ>= `  M ) )
31, 2syl 16 . 2  |-  ( ph  ->  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
4 seqfn 11264 . . 3  |-  ( M  e.  ZZ  ->  seq  M ( Q ,  F
)  Fn  ( ZZ>= `  M ) )
51, 4syl 16 . 2  |-  ( ph  ->  seq  M ( Q ,  F )  Fn  ( ZZ>= `  M )
)
6 simpr 448 . . 3  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  a  e.  ( ZZ>= `  M )
)
7 simpll 731 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  x  e.  ( M ... a ) )  ->  ph )
8 elfzuz 10989 . . . . 5  |-  ( x  e.  ( M ... a )  ->  x  e.  ( ZZ>= `  M )
)
98adantl 453 . . . 4  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  x  e.  ( M ... a ) )  ->  x  e.  ( ZZ>= `  M )
)
10 seqfeq3.f . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
117, 9, 10syl2anc 643 . . 3  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  x  e.  ( M ... a ) )  ->  ( F `  x )  e.  S
)
12 seqfeq3.cl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
1312adantlr 696 . . 3  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
14 seqfeq3.id . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
1514adantlr 696 . . 3  |-  ( ( ( ph  /\  a  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
166, 11, 13, 15seqfeq4 11301 . 2  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  .+  ,  F
) `  a )  =  (  seq  M ( Q ,  F ) `
 a ) )
173, 5, 16eqfnfvd 5771 1  |-  ( ph  ->  seq  M (  .+  ,  F )  =  seq  M ( Q ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    Fn wfn 5391   ` cfv 5396  (class class class)co 6022   ZZcz 10216   ZZ>=cuz 10422   ...cfz 10977    seq cseq 11252
This theorem is referenced by:  mulgpropd  14852  esumfsupre  24259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-seq 11253
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