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Metamath Proof Explorer

Theorem ressmulgnn

Description: Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017)

Ref Expression
Hypotheses ressmulgnn.1 
|- H = ( G |`s A )
ressmulgnn.2 
|- A C_ ( Base ` G )
ressmulgnn.3 
|- .* = ( .g ` G )
ressmulgnn.4 
|- I = ( invg ` G )
Assertion ressmulgnn 
|- ( ( N e. NN /\ X e. A ) -> ( N ( .g ` H ) X ) = ( N .* X ) )

Proof

Step Hyp Ref Expression
1 ressmulgnn.1 
 |-  H = ( G |`s A )
2 ressmulgnn.2 
 |-  A C_ ( Base ` G )
3 ressmulgnn.3 
 |-  .* = ( .g ` G )
4 ressmulgnn.4 
 |-  I = ( invg ` G )
5 eqid 
 |-  ( Base ` G ) = ( Base ` G )
6 1 5 ressbas2 
 |-  ( A C_ ( Base ` G ) -> A = ( Base ` H ) )
7 2 6 ax-mp 
 |-  A = ( Base ` H )
8 eqid 
 |-  ( +g ` H ) = ( +g ` H )
9 eqid 
 |-  ( .g ` H ) = ( .g ` H )
10 fvex 
 |-  ( Base ` G ) e. _V
11 10 2 ssexi 
 |-  A e. _V
12 eqid 
 |-  ( +g ` G ) = ( +g ` G )
13 1 12 ressplusg 
 |-  ( A e. _V -> ( +g ` G ) = ( +g ` H ) )
14 11 13 ax-mp 
 |-  ( +g ` G ) = ( +g ` H )
15 seqeq2 
 |-  ( ( +g ` G ) = ( +g ` H ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
16 14 15 ax-mp 
 |-  seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
17 7 8 9 16 mulgnn 
 |-  ( ( N e. NN /\ X e. A ) -> ( N ( .g ` H ) X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
18 simpr 
 |-  ( ( N e. NN /\ X e. A ) -> X e. A )
19 2 18 sselid 
 |-  ( ( N e. NN /\ X e. A ) -> X e. ( Base ` G ) )
20 eqid 
 |-  seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
21 5 12 3 20 mulgnn 
 |-  ( ( N e. NN /\ X e. ( Base ` G ) ) -> ( N .* X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
22 19 21 syldan 
 |-  ( ( N e. NN /\ X e. A ) -> ( N .* X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
23 17 22 eqtr4d 
 |-  ( ( N e. NN /\ X e. A ) -> ( N ( .g ` H ) X ) = ( N .* X ) )