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Theorem ressmulgnn0 24159
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  =  ( inv g `  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 448 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  e.  NN )  ->  N  e.  NN )
2 simplr 732 . . 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  =  ( inv g `  G )
73, 4, 5, 6ressmulgnn 24158 . . 3  |-  ( ( N  e.  NN  /\  X  e.  A )  ->  ( N (.g `  H
) X )  =  ( N  .*  X
) )
81, 2, 7syl2anc 643 . 2  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  e.  NN )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
9 simplr 732 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  X  e.  A )
10 eqid 2404 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
113, 10ressbas2 13475 . . . . . . 7  |-  ( A 
C_  ( Base `  G
)  ->  A  =  ( Base `  H )
)
124, 11ax-mp 8 . . . . . 6  |-  A  =  ( Base `  H
)
13 ressmulgnn0.4 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  H
)
14 eqid 2404 . . . . . 6  |-  (.g `  H
)  =  (.g `  H
)
1512, 13, 14mulg0 14850 . . . . 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 448 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  N  = 
0 )
1817oveq1d 6055 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N
(.g `  H ) X )  =  ( 0 (.g `  H ) X ) )
194, 9sseldi 3306 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  X  e.  ( Base `  G )
)
20 eqid 2404 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
2110, 20, 5mulg0 14850 . . . . 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 2446 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N
(.g `  H ) X )  =  ( 0  .*  X ) )
2417oveq1d 6055 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N  .*  X )  =  ( 0  .*  X ) )
2523, 24eqtr4d 2439 . 2  |-  ( ( ( N  e.  NN0  /\  X  e.  A )  /\  N  =  0 )  ->  ( N
(.g `  H ) X )  =  ( N  .*  X ) )
26 elnn0 10179 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2726biimpi 187 . . 3  |-  ( N  e.  NN0  ->  ( N  e.  NN  \/  N  =  0 ) )
2827adantr 452 . 2  |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N  e.  NN  \/  N  =  0
) )
298, 25, 28mpjaodan 762 1  |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N (.g `  H
) X )  =  ( N  .*  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   ` cfv 5413  (class class class)co 6040   0cc0 8946   NNcn 9956   NN0cn0 10177   Basecbs 13424   ↾s cress 13425   0gc0g 13678   inv gcminusg 14641  .gcmg 14644
This theorem is referenced by:  xrge0mulgnn0  24163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-seq 11279  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulg 14770
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