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Theorem gsumnunsn 27071
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 11922 . . 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 11009 . . . 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 2539 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  /\  k  =  ( P  + 
1 ) )  ->  B  e.  K )
16 elfzp1 11606 . . . . . . 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 784 . . . 4  |-  ( (
ph  /\  k  e.  ( N ... ( P  +  1 ) ) )  ->  B  e.  K )
20 eqid 2451 . . . 4  |-  ( k  e.  ( N ... ( P  +  1
) )  |->  B )  =  ( k  e.  ( N ... ( P  +  1 ) )  |->  B )
2119, 20fmptd 5966 . . 3  |-  ( ph  ->  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) : ( N ... ( P  +  1 ) ) --> K )
224, 5, 6, 8, 21gsumval2 15615 . 2  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  ( P  +  1 ) ) )
23 eqid 2451 . . . . . 6  |-  ( k  e.  ( N ... P )  |->  B )  =  ( k  e.  ( N ... P
)  |->  B )
249, 23fmptd 5966 . . . . 5  |-  ( ph  ->  ( k  e.  ( N ... P ) 
|->  B ) : ( N ... P ) --> K )
254, 5, 6, 1, 24gsumval2 15615 . . . 4  |-  ( ph  ->  ( M  gsumg  ( k  e.  ( N ... P ) 
|->  B ) )  =  (  seq N ( 
.+  ,  ( k  e.  ( N ... P )  |->  B ) ) `  P ) )
26 fzssp1 11602 . . . . . . . 8  |-  ( N ... P )  C_  ( N ... ( P  +  1 ) )
27 resmpt 5254 . . . . . . . 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 5790 . . . . . 6  |-  ( ( ( k  e.  ( N ... ( P  +  1 ) ) 
|->  B )  |`  ( N ... P ) ) `
 i )  =  ( ( k  e.  ( N ... P
)  |->  B ) `  i )
30 fvres 5803 . . . . . . 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 2507 . . . . 5  |-  ( (
ph  /\  i  e.  ( N ... P ) )  ->  ( (
k  e.  ( N ... ( P  + 
1 ) )  |->  B ) `  i )  =  ( ( k  e.  ( N ... P )  |->  B ) `
 i ) )
331, 32seqfveq 11931 . . . 4  |-  ( ph  ->  (  seq N ( 
.+  ,  ( k  e.  ( N ... ( P  +  1
) )  |->  B ) ) `  P )  =  (  seq N
(  .+  ,  (
k  e.  ( N ... P )  |->  B ) ) `  P
) )
3425, 33eqtr4d 2495 . . 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 11560 . . . . . 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 5878 . . . 4  |-  ( ph  ->  ( ( k  e.  ( N ... ( P  +  1 ) )  |->  B ) `  ( P  +  1
) )  =  C )
3938eqcomd 2459 . . 3  |-  ( ph  ->  C  =  ( ( k  e.  ( N ... ( P  + 
1 ) )  |->  B ) `  ( P  +  1 ) ) )
4034, 39oveq12d 6208 . 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 2502 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 1370    e. wcel 1758    C_ wss 3426    |-> cmpt 4448    |` cres 4940   ` cfv 5516  (class class class)co 6190   1c1 9384    + caddc 9386   ZZ>=cuz 10962   ...cfz 11538    seqcseq 11907   Basecbs 14276   +g cplusg 14340    gsumg cgsu 14481   Mndcmnd 15511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-seq 11908  df-0g 14482  df-gsum 14483
This theorem is referenced by:  signstfvn  27104
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