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Theorem fincmpzer 25453
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 3210 . . . . 5  |-  ( M ... N )  C_  ( M ... N )
2 eqid 2296 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  U )  =  ( k  e.  ( M ... N
)  |->  U )
32prodeqfv 25421 . . . . 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 25423 . . . 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 2342 . . 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 2296 . . . . . . 7  |-  ran  G  =  ran  G
9 fincmpzer.1 . . . . . . 7  |-  U  =  (GId `  G )
108, 9iorlid 21011 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  ran  G )
118, 9cmpidelt 21012 . . . . . 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 2297 . . . . 5  |-  ( k  =  x  ->  U  =  U )
17 fvex 5555 . . . . . 6  |-  (GId `  G )  e.  _V
189, 17eqeltri 2366 . . . . 5  |-  U  e. 
_V
1916, 2, 18fvmpt 5618 . . . 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 11106 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )
)  ->  (  seq  M ( G ,  ( k  e.  ( M ... N )  |->  U ) ) `  N
)  =  U )
227, 21eqtrd 2328 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062  GIdcgi 20870    ExId cexid 20997   Magmacmagm 21001   prod_cprd 25401
This theorem is referenced by:  svli2  25587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-gid 20875  df-exid 20998  df-mgm 21002  df-prod 25402
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