Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gsumnunsn Structured version   Unicode version

Theorem gsumnunsn 26867
Description: Closure of a group sum in a non-commutative monoid. (Contributed by Thierry Arnoux, 8-Oct-2018.)
Hypotheses
Ref Expression
gsumncl.k  |-  K  =  ( Base `  M
)
gsumncl.w  |-  ( ph  ->  M  e.  Mnd )
gsumncl.p  |-  ( ph  ->  P  e.  ( ZZ>= `  N ) )
gsumncl.b  |-  ( (
ph  /\  k  e.  ( N ... P ) )  ->  B  e.  K )
gsumnunsn.a  |-  .+  =  ( +g  `  M )
gsumnunsn.l  |-  ( ph  ->  C  e.  K )
gsumnunsn.c  |-  ( (
ph  /\  k  =  ( P  +  1
) )  ->  B  =  C )
Assertion
Ref Expression
gsumnunsn  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) )  =  ( ( M  gsumg  ( k  e.  ( N ... P )  |->  B ) )  .+  C ) )
Distinct variable groups:    k, K    k, N    P, k    ph, k    C, k
Allowed substitution hints:    B( k)    .+ ( k)    M( k)

Proof of Theorem gsumnunsn
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 gsumncl.p . . 3  |-  ( ph  ->  P  e.  ( ZZ>= `  N ) )
2 seqp1 11817 . . 3  |-  ( P  e.  ( ZZ>= `  N
)  ->  (  seq N (  .+  , 
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) ) `  ( P  +  1
) )  =  ( (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  P ) 
.+  ( ( k  e.  ( N ... ( P  +  1
) )  |->  B ) `
 ( P  + 
1 ) ) ) )
31, 2syl 16 . 2  |-  ( ph  ->  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  ( P  +  1 ) )  =  ( (  seq N (  .+  , 
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) ) `  P )  .+  (
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) `  ( P  +  1 ) ) ) )
4 gsumncl.k . . 3  |-  K  =  ( Base `  M
)
5 gsumnunsn.a . . 3  |-  .+  =  ( +g  `  M )
6 gsumncl.w . . 3  |-  ( ph  ->  M  e.  Mnd )
7 peano2uz 10904 . . . 4  |-  ( P  e.  ( ZZ>= `  N
)  ->  ( P  +  1 )  e.  ( ZZ>= `  N )
)
81, 7syl 16 . . 3  |-  ( ph  ->  ( P  +  1 )  e.  ( ZZ>= `  N ) )
9 gsumncl.b . . . . . 6  |-  ( (
ph  /\  k  e.  ( N ... P ) )  ->  B  e.  K )
109adantlr 709 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  e.  ( N ... P
) )  ->  B  e.  K )
11 gsumnunsn.c . . . . . . 7  |-  ( (
ph  /\  k  =  ( P  +  1
) )  ->  B  =  C )
1211adantlr 709 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  =  ( P  + 
1 ) )  ->  B  =  C )
13 gsumnunsn.l . . . . . . 7  |-  ( ph  ->  C  e.  K )
1413ad2antrr 720 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  =  ( P  + 
1 ) )  ->  C  e.  K )
1512, 14eqeltrd 2515 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  =  ( P  + 
1 ) )  ->  B  e.  K )
16 elfzp1 11501 . . . . . . 7  |-  ( P  e.  ( ZZ>= `  N
)  ->  ( k  e.  ( N ... ( P  +  1 ) )  <->  ( k  e.  ( N ... P
)  \/  k  =  ( P  +  1 ) ) ) )
171, 16syl 16 . . . . . 6  |-  ( ph  ->  ( k  e.  ( N ... ( P  +  1 ) )  <-> 
( k  e.  ( N ... P )  \/  k  =  ( P  +  1 ) ) ) )
1817biimpa 481 . . . . 5  |-  ( (
ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  ->  ( k  e.  ( N ... P
)  \/  k  =  ( P  +  1 ) ) )
1910, 15, 18mpjaodan 779 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  ->  B  e.  K )
20 eqid 2441 . . . 4  |-  ( k  e.  ( N ... ( P  +  1
) )  |->  B )  =  ( k  e.  ( N ... ( P  +  1 ) )  |->  B )
2119, 20fmptd 5864 . . 3  |-  ( ph  ->  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) : ( N ... ( P  +  1 ) ) --> K )
224, 5, 6, 8, 21gsumval2 15506 . 2  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  ( P  +  1 ) ) )
23 eqid 2441 . . . . . 6  |-  ( k  e.  ( N ... P )  |->  B )  =  ( k  e.  ( N ... P
)  |->  B )
249, 23fmptd 5864 . . . . 5  |-  ( ph  ->  ( k  e.  ( N ... P ) 
|->  B ) : ( N ... P ) --> K )
254, 5, 6, 1, 24gsumval2 15506 . . . 4  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... P ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... P )  |->  B ) ) `  P ) )
26 fzssp1 11497 . . . . . . . 8  |-  ( N ... P )  C_  ( N ... ( P  +  1 ) )
27 resmpt 5153 . . . . . . . 8  |-  ( ( N ... P ) 
C_  ( N ... ( P  +  1
) )  ->  (
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  |`  ( N ... P ) )  =  ( k  e.  ( N ... P
)  |->  B ) )
2826, 27ax-mp 5 . . . . . . 7  |-  ( ( k  e.  ( N ... ( P  + 
1 ) )  |->  B )  |`  ( N ... P ) )  =  ( k  e.  ( N ... P ) 
|->  B )
2928fveq1i 5689 . . . . . 6  |-  ( ( ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  |`  ( N ... P ) ) `
 i )  =  ( ( k  e.  ( N ... P
)  |->  B ) `  i )
30 fvres 5701 . . . . . . 7  |-  ( i  e.  ( N ... P )  ->  (
( ( k  e.  ( N ... ( P  +  1 ) )  |->  B )  |`  ( N ... P ) ) `  i )  =  ( ( k  e.  ( N ... ( P  +  1
) )  |->  B ) `
 i ) )
3130adantl 463 . . . . . 6  |-  ( (
ph  /\  i  e.  ( N ... P ) )  ->  ( (
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  |`  ( N ... P ) ) `
 i )  =  ( ( k  e.  ( N ... ( P  +  1 ) )  |->  B ) `  i ) )
3229, 31syl5reqr 2488 . . . . 5  |-  ( (
ph  /\  i  e.  ( N ... P ) )  ->  ( (
k  e.  ( N ... ( P  + 
1 ) )  |->  B ) `  i )  =  ( ( k  e.  ( N ... P )  |->  B ) `
 i ) )
331, 32seqfveq 11826 . . . 4  |-  ( ph  ->  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  P )  =  (  seq N
(  .+  ,  (
k  e.  ( N ... P )  |->  B ) ) `  P
) )
3425, 33eqtr4d 2476 . . 3  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... P ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  P ) )
3520a1i 11 . . . . 5  |-  ( ph  ->  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  =  ( k  e.  ( N ... ( P  + 
1 ) )  |->  B ) )
36 eluzfz2 11455 . . . . . 6  |-  ( ( P  +  1 )  e.  ( ZZ>= `  N
)  ->  ( P  +  1 )  e.  ( N ... ( P  +  1 ) ) )
378, 36syl 16 . . . . 5  |-  ( ph  ->  ( P  +  1 )  e.  ( N ... ( P  + 
1 ) ) )
3835, 11, 37, 13fvmptd 5776 . . . 4  |-  ( ph  ->  ( ( k  e.  ( N ... ( P  +  1 ) )  |->  B ) `  ( P  +  1
) )  =  C )
3938eqcomd 2446 . . 3  |-  ( ph  ->  C  =  ( ( k  e.  ( N ... ( P  + 
1 ) )  |->  B ) `  ( P  +  1 ) ) )
4034, 39oveq12d 6108 . 2  |-  ( ph  ->  ( ( M  gsumg  ( k  e.  ( N ... P )  |->  B ) )  .+  C )  =  ( (  seq N (  .+  , 
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) ) `  P )  .+  (
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) `  ( P  +  1 ) ) ) )
413, 22, 403eqtr4d 2483 1  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) )  =  ( ( M  gsumg  ( k  e.  ( N ... P )  |->  B ) )  .+  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    C_ wss 3325    e. cmpt 4347    |` cres 4838   ` cfv 5415  (class class class)co 6090   1c1 9279    + caddc 9281   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802   Basecbs 14170   +g cplusg 14234    gsumg cgsu 14375   Mndcmnd 15405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-0g 14376  df-gsum 14377
This theorem is referenced by:  signstfvn  26900
  Copyright terms: Public domain W3C validator