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Theorem ressmulgnn0 26062
Description: Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 14-Jun-2017.)
Hypotheses
Ref Expression
ressmulgnn.1  |-  H  =  ( Gs  A )
ressmulgnn.2  |-  A  C_  ( Base `  G )
ressmulgnn.3  |-  .*  =  (.g
`  G )
ressmulgnn.4  |-  I  =  ( invg `  G )
ressmulgnn0.4  |-  ( 0g
`  G )  =  ( 0g `  H
)
Assertion
Ref Expression
ressmulgnn0  |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N (.g `  H
) X )  =  ( N  .*  X
) )

Proof of Theorem ressmulgnn0
StepHypRef Expression
1 simpr 458 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  e.  NN )  ->  N  e.  NN )
2 simplr 749 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  e.  NN )  ->  X  e.  A
)
3 ressmulgnn.1 . . . 4  |-  H  =  ( Gs  A )
4 ressmulgnn.2 . . . 4  |-  A  C_  ( Base `  G )
5 ressmulgnn.3 . . . 4  |-  .*  =  (.g
`  G )
6 ressmulgnn.4 . . . 4  |-  I  =  ( invg `  G )
73, 4, 5, 6ressmulgnn 26061 . . 3  |-  ( ( N  e.  NN  /\  X  e.  A )  ->  ( N (.g `  H
) X )  =  ( N  .*  X
) )
81, 2, 7syl2anc 656 . 2  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  e.  NN )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
9 simplr 749 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  X  e.  A )
10 eqid 2441 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
113, 10ressbas2 14225 . . . . . . 7  |-  ( A 
C_  ( Base `  G
)  ->  A  =  ( Base `  H )
)
124, 11ax-mp 5 . . . . . 6  |-  A  =  ( Base `  H
)
13 ressmulgnn0.4 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  H
)
14 eqid 2441 . . . . . 6  |-  (.g `  H
)  =  (.g `  H
)
1512, 13, 14mulg0 15625 . . . . 5  |-  ( X  e.  A  ->  (
0 (.g `  H ) X )  =  ( 0g
`  G ) )
169, 15syl 16 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( 0 (.g `  H ) X )  =  ( 0g
`  G ) )
17 simpr 458 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  N  = 
0 )
1817oveq1d 6105 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N
(.g `  H ) X )  =  ( 0 (.g `  H ) X ) )
194, 9sseldi 3351 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  X  e.  ( Base `  G )
)
20 eqid 2441 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
2110, 20, 5mulg0 15625 . . . . 5  |-  ( X  e.  ( Base `  G
)  ->  ( 0  .*  X )  =  ( 0g `  G
) )
2219, 21syl 16 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( 0  .*  X )  =  ( 0g `  G
) )
2316, 18, 223eqtr4d 2483 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N
(.g `  H ) X )  =  ( 0  .*  X ) )
2417oveq1d 6105 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N  .*  X )  =  ( 0  .*  X ) )
2523, 24eqtr4d 2476 . 2  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N
(.g `  H ) X )  =  ( N  .*  X ) )
26 elnn0 10577 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2726biimpi 194 . . 3  |-  ( N  e.  NN0  ->  ( N  e.  NN  \/  N  =  0 ) )
2827adantr 462 . 2  |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N  e.  NN  \/  N  =  0
) )
298, 25, 28mpjaodan 779 1  |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N (.g `  H
) X )  =  ( N  .*  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    C_ wss 3325   ` cfv 5415  (class class class)co 6090   0cc0 9278   NNcn 10318   NN0cn0 10575   Basecbs 14170   ↾s cress 14171   0gc0g 14374   invgcminusg 15407  .gcmg 15410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-seq 11803  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulg 15541
This theorem is referenced by:  xrge0mulgnn0  26067
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