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Theorem fincmpzer 25350
Description: Finite composite of identity elements. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2014.)
Hypothesis
Ref Expression
fincmpzer.1  |-  U  =  (GId `  G )
Assertion
Ref Expression
fincmpzer  |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )
)  ->  prod_ k  e.  ( M ... N
) G U  =  U )
Distinct variable groups:    k, M    k, N    U, k
Allowed substitution hint:    G( k)

Proof of Theorem fincmpzer
Dummy variables  x  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3197 . . . . 5  |-  ( M ... N )  C_  ( M ... N )
2 eqid 2283 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  U )  =  ( k  e.  ( M ... N
)  |->  U )
32prodeqfv 25318 . . . . 5  |-  ( ( M ... N ) 
C_  ( M ... N )  ->  prod_ m  e.  ( M ... N ) G ( ( k  e.  ( M ... N ) 
|->  U ) `  m
)  =  prod_ k  e.  ( M ... N
) G U )
41, 3ax-mp 8 . . . 4  |-  prod_ m  e.  ( M ... N
) G ( ( k  e.  ( M ... N )  |->  U ) `  m )  =  prod_ k  e.  ( M ... N ) G U
5 fprodser 25320 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  prod_ m  e.  ( M ... N
) G ( ( k  e.  ( M ... N )  |->  U ) `  m )  =  (  seq  M
( G ,  ( k  e.  ( M ... N )  |->  U ) ) `  N
) )
64, 5syl5eqr 2329 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  prod_ k  e.  ( M ... N
) G U  =  (  seq  M ( G ,  ( k  e.  ( M ... N )  |->  U ) ) `  N ) )
76adantr 451 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )
)  ->  prod_ k  e.  ( M ... N
) G U  =  (  seq  M ( G ,  ( k  e.  ( M ... N )  |->  U ) ) `  N ) )
8 eqid 2283 . . . . . . 7  |-  ran  G  =  ran  G
9 fincmpzer.1 . . . . . . 7  |-  U  =  (GId `  G )
108, 9iorlid 20995 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  ran  G )
118, 9cmpidelt 20996 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  U  e. 
ran  G )  -> 
( ( U G U )  =  U  /\  ( U G U )  =  U ) )
1210, 11mpdan 649 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( ( U G U )  =  U  /\  ( U G U )  =  U ) )
1312simpld 445 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( U G U )  =  U )
1413adantl 452 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )
)  ->  ( U G U )  =  U )
15 simpl 443 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )
)  ->  N  e.  ( ZZ>= `  M )
)
16 eqidd 2284 . . . . 5  |-  ( k  =  x  ->  U  =  U )
17 fvex 5539 . . . . . 6  |-  (GId `  G )  e.  _V
189, 17eqeltri 2353 . . . . 5  |-  U  e. 
_V
1916, 2, 18fvmpt 5602 . . . 4  |-  ( x  e.  ( M ... N )  ->  (
( k  e.  ( M ... N ) 
|->  U ) `  x
)  =  U )
2019adantl 452 . . 3  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  G  e.  ( Magma  i^i 
ExId  ) )  /\  x  e.  ( M ... N ) )  -> 
( ( k  e.  ( M ... N
)  |->  U ) `  x )  =  U )
2114, 15, 20seqid3 11090 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )
)  ->  (  seq  M ( G ,  ( k  e.  ( M ... N )  |->  U ) ) `  N
)  =  U )
227, 21eqtrd 2315 1  |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )
)  ->  prod_ k  e.  ( M ... N
) G U  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046  GIdcgi 20854    ExId cexid 20981   Magmacmagm 20985   prod_cprd 25298
This theorem is referenced by:  svli2  25484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-gid 20859  df-exid 20982  df-mgm 20986  df-prod 25299
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