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Theorem gsumnunsn 28366
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 12101 . . 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 11143 . . . 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 714 . . . . 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 714 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  =  ( P  + 
1 ) )  ->  B  =  C )
13 gsumnunsn.l . . . . . . 7  |-  ( ph  ->  C  e.  K )
1413ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  =  ( P  + 
1 ) )  ->  C  e.  K )
1512, 14eqeltrd 2531 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  =  ( P  + 
1 ) )  ->  B  e.  K )
16 elfzp1 11739 . . . . . . 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 484 . . . . 5  |-  ( (
ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  ->  ( k  e.  ( N ... P
)  \/  k  =  ( P  +  1 ) ) )
1910, 15, 18mpjaodan 786 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  ->  B  e.  K )
20 eqid 2443 . . . 4  |-  ( k  e.  ( N ... ( P  +  1
) )  |->  B )  =  ( k  e.  ( N ... ( P  +  1 ) )  |->  B )
2119, 20fmptd 6040 . . 3  |-  ( ph  ->  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) : ( N ... ( P  +  1 ) ) --> K )
224, 5, 6, 8, 21gsumval2 15781 . 2  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  ( P  +  1 ) ) )
23 eqid 2443 . . . . . 6  |-  ( k  e.  ( N ... P )  |->  B )  =  ( k  e.  ( N ... P
)  |->  B )
249, 23fmptd 6040 . . . . 5  |-  ( ph  ->  ( k  e.  ( N ... P ) 
|->  B ) : ( N ... P ) --> K )
254, 5, 6, 1, 24gsumval2 15781 . . . 4  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... P ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... P )  |->  B ) ) `  P ) )
26 fzssp1 11735 . . . . . . . 8  |-  ( N ... P )  C_  ( N ... ( P  +  1 ) )
27 resmpt 5313 . . . . . . . 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 5857 . . . . . 6  |-  ( ( ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  |`  ( N ... P ) ) `
 i )  =  ( ( k  e.  ( N ... P
)  |->  B ) `  i )
30 fvres 5870 . . . . . . 7  |-  ( i  e.  ( N ... P )  ->  (
( ( k  e.  ( N ... ( P  +  1 ) )  |->  B )  |`  ( N ... P ) ) `  i )  =  ( ( k  e.  ( N ... ( P  +  1
) )  |->  B ) `
 i ) )
3130adantl 466 . . . . . 6  |-  ( (
ph  /\  i  e.  ( N ... P ) )  ->  ( (
( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  |`  ( N ... P ) ) `
 i )  =  ( ( k  e.  ( N ... ( P  +  1 ) )  |->  B ) `  i ) )
3229, 31syl5reqr 2499 . . . . 5  |-  ( (
ph  /\  i  e.  ( N ... P ) )  ->  ( (
k  e.  ( N ... ( P  + 
1 ) )  |->  B ) `  i )  =  ( ( k  e.  ( N ... P )  |->  B ) `
 i ) )
331, 32seqfveq 12110 . . . 4  |-  ( ph  ->  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  P )  =  (  seq N
(  .+  ,  (
k  e.  ( N ... P )  |->  B ) ) `  P
) )
3425, 33eqtr4d 2487 . . 3  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... P ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  P ) )
35 eqidd 2444 . . . . 5  |-  ( ph  ->  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  =  ( k  e.  ( N ... ( P  + 
1 ) )  |->  B ) )
36 eluzfz2 11703 . . . . . 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 5946 . . . 4  |-  ( ph  ->  ( ( k  e.  ( N ... ( P  +  1 ) )  |->  B ) `  ( P  +  1
) )  =  C )
3938eqcomd 2451 . . 3  |-  ( ph  ->  C  =  ( ( k  e.  ( N ... ( P  + 
1 ) )  |->  B ) `  ( P  +  1 ) ) )
4034, 39oveq12d 6299 . 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 2494 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 1383    e. wcel 1804    C_ wss 3461    |-> cmpt 4495    |` cres 4991   ` cfv 5578  (class class class)co 6281   1c1 9496    + caddc 9498   ZZ>=cuz 11090   ...cfz 11681    seqcseq 12086   Basecbs 14509   +g cplusg 14574    gsumg cgsu 14715   Mndcmnd 15793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-seq 12087  df-0g 14716  df-gsum 14717
This theorem is referenced by:  signstfvn  28399
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