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Theorem seqfeq3 11363
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 11325 . . 3  |-  ( M  e.  ZZ  ->  seq  M (  .+  ,  F
)  Fn  ( ZZ>= `  M ) )
31, 2syl 16 . 2  |-  ( ph  ->  seq  M (  .+  ,  F )  Fn  ( ZZ>=
`  M ) )
4 seqfn 11325 . . 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 11045 . . . . 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 11362 . 2  |-  ( (
ph  /\  a  e.  ( ZZ>= `  M )
)  ->  (  seq  M (  .+  ,  F
) `  a )  =  (  seq  M ( Q ,  F ) `
 a ) )
173, 5, 16eqfnfvd 5822 1  |-  ( ph  ->  seq  M (  .+  ,  F )  =  seq  M ( Q ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   ZZcz 10272   ZZ>=cuz 10478   ...cfz 11033    seq cseq 11313
This theorem is referenced by:  mulgpropd  14913  esumfsupre  24451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-seq 11314
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