ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  gsumfzsubmcl Unicode version

Theorem gsumfzsubmcl 13408
Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
Hypotheses
Ref Expression
gsumfzsubmcl.g  |-  ( ph  ->  G  e.  Mnd )
gsumfzsubmcl.m  |-  ( ph  ->  M  e.  ZZ )
gsumfzsubmcl.n  |-  ( ph  ->  N  e.  ZZ )
gsumsubmcl.s  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
gsumfzsubmcl.f  |-  ( ph  ->  F : ( M ... N ) --> S )
Assertion
Ref Expression
gsumfzsubmcl  |-  ( ph  ->  ( G  gsumg  F )  e.  S
)

Proof of Theorem gsumfzsubmcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2193 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2193 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2193 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
4 gsumfzsubmcl.g . . . . . 6  |-  ( ph  ->  G  e.  Mnd )
5 gsumfzsubmcl.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
6 gsumfzsubmcl.n . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
7 gsumfzsubmcl.f . . . . . . 7  |-  ( ph  ->  F : ( M ... N ) --> S )
8 gsumsubmcl.s . . . . . . . 8  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
91submss 13048 . . . . . . . 8  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  ( Base `  G ) )
108, 9syl 14 . . . . . . 7  |-  ( ph  ->  S  C_  ( Base `  G ) )
117, 10fssd 5416 . . . . . 6  |-  ( ph  ->  F : ( M ... N ) --> (
Base `  G )
)
121, 2, 3, 4, 5, 6, 11gsumfzval 12974 . . . . 5  |-  ( ph  ->  ( G  gsumg  F )  =  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M ( ( +g  `  G ) ,  F ) `  N ) ) )
1312adantr 276 . . . 4  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  F )  =  if ( N  <  M , 
( 0g `  G
) ,  (  seq M ( ( +g  `  G ) ,  F
) `  N )
) )
14 simpr 110 . . . . 5  |-  ( (
ph  /\  N  <  M )  ->  N  <  M )
1514iftrued 3564 . . . 4  |-  ( (
ph  /\  N  <  M )  ->  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M
( ( +g  `  G
) ,  F ) `
 N ) )  =  ( 0g `  G ) )
1613, 15eqtrd 2226 . . 3  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  F )  =  ( 0g
`  G ) )
172subm0cl 13050 . . . . 5  |-  ( S  e.  (SubMnd `  G
)  ->  ( 0g `  G )  e.  S
)
188, 17syl 14 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  S )
1918adantr 276 . . 3  |-  ( (
ph  /\  N  <  M )  ->  ( 0g `  G )  e.  S
)
2016, 19eqeltrd 2270 . 2  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  F )  e.  S )
2112adantr 276 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  F )  =  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M ( ( +g  `  G ) ,  F ) `  N ) ) )
22 simpr 110 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  -.  N  <  M )
2322iffalsed 3567 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  if ( N  <  M , 
( 0g `  G
) ,  (  seq M ( ( +g  `  G ) ,  F
) `  N )
)  =  (  seq M ( ( +g  `  G ) ,  F
) `  N )
)
2421, 23eqtrd 2226 . . 3  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  F )  =  (  seq M ( ( +g  `  G ) ,  F ) `  N ) )
255adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  M  e.  ZZ )
266adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  N  e.  ZZ )
2725zred 9439 . . . . . 6  |-  ( (
ph  /\  -.  N  <  M )  ->  M  e.  RR )
2826zred 9439 . . . . . 6  |-  ( (
ph  /\  -.  N  <  M )  ->  N  e.  RR )
2927, 28, 22nltled 8140 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  M  <_  N )
30 eluz2 9598 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
3125, 26, 29, 30syl3anbrc 1183 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  N  e.  ( ZZ>= `  M )
)
327adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  F : ( M ... N ) --> S )
3332ffvelcdmda 5693 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  x  e.  ( M ... N ) )  -> 
( F `  x
)  e.  S )
348ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  S  e.  (SubMnd `  G )
)
35 simprl 529 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  x  e.  S )
36 simprr 531 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  y  e.  S )
373submcl 13051 . . . . 5  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x ( +g  `  G
) y )  e.  S )
3834, 35, 36, 37syl3anc 1249 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x ( +g  `  G
) y )  e.  S )
395, 6fzfigd 10502 . . . . . 6  |-  ( ph  ->  ( M ... N
)  e.  Fin )
4039adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  ( M ... N )  e. 
Fin )
4132, 40fexd 5788 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  F  e.  _V )
42 plusgslid 12730 . . . . . . 7  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
4342slotex 12645 . . . . . 6  |-  ( G  e.  Mnd  ->  ( +g  `  G )  e. 
_V )
444, 43syl 14 . . . . 5  |-  ( ph  ->  ( +g  `  G
)  e.  _V )
4544adantr 276 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  ( +g  `  G )  e. 
_V )
4631, 33, 38, 41, 45seqclg 10543 . . 3  |-  ( (
ph  /\  -.  N  <  M )  ->  (  seq M ( ( +g  `  G ) ,  F
) `  N )  e.  S )
4724, 46eqeltrd 2270 . 2  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  F )  e.  S
)
48 zdclt 9394 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  -> DECID  N  <  M )
496, 5, 48syl2anc 411 . . 3  |-  ( ph  -> DECID  N  <  M )
50 exmiddc 837 . . 3  |-  (DECID  N  < 
M  ->  ( N  <  M  \/  -.  N  <  M ) )
5149, 50syl 14 . 2  |-  ( ph  ->  ( N  <  M  \/  -.  N  <  M
) )
5220, 47, 51mpjaodan 799 1  |-  ( ph  ->  ( G  gsumg  F )  e.  S
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2164   _Vcvv 2760    C_ wss 3153   ifcif 3557   class class class wbr 4029   -->wf 5250   ` cfv 5254  (class class class)co 5918   Fincfn 6794    < clt 8054    <_ cle 8055   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074    seqcseq 10518   Basecbs 12618   +g cplusg 12695   0gc0g 12867    gsumg cgsu 12868   Mndcmnd 12997  SubMndcsubmnd 13030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870  df-submnd 13032
This theorem is referenced by:  lgseisenlem3  15188
  Copyright terms: Public domain W3C validator