Metamath Proof Explorer

Theorem ressmulgnn0

Description: Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-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 )
ressmulgnn0.4 
|- ( 0g ` G ) = ( 0g ` H )
Assertion ressmulgnn0 
|- ( ( N e. NN0 /\ 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 ressmulgnn0.4 
 |-  ( 0g ` G ) = ( 0g ` H )
6 simpr 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N e. NN ) -> N e. NN )
7 simplr 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N e. NN ) -> X e. A )
8 1 2 3 4 ressmulgnn 
 |-  ( ( N e. NN /\ X e. A ) -> ( N ( .g ` H ) X ) = ( N .* X ) )
9 6 7 8 syl2anc 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N e. NN ) -> ( N ( .g ` H ) X ) = ( N .* X ) )
10 simplr 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> X e. A )
11 eqid 
 |-  ( Base ` G ) = ( Base ` G )
12 1 11 ressbas2 
 |-  ( A C_ ( Base ` G ) -> A = ( Base ` H ) )
13 2 12 ax-mp 
 |-  A = ( Base ` H )
14 eqid 
 |-  ( .g ` H ) = ( .g ` H )
15 13 5 14 mulg0 
 |-  ( X e. A -> ( 0 ( .g ` H ) X ) = ( 0g ` G ) )
16 10 15 syl 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> ( 0 ( .g ` H ) X ) = ( 0g ` G ) )
17 simpr 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> N = 0 )
18 17 oveq1d 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> ( N ( .g ` H ) X ) = ( 0 ( .g ` H ) X ) )
19 2 10 sselid 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> X e. ( Base ` G ) )
20 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
21 11 20 3 mulg0 
 |-  ( X e. ( Base ` G ) -> ( 0 .* X ) = ( 0g ` G ) )
22 19 21 syl 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> ( 0 .* X ) = ( 0g ` G ) )
23 16 18 22 3eqtr4d 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> ( N ( .g ` H ) X ) = ( 0 .* X ) )
24 17 oveq1d 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> ( N .* X ) = ( 0 .* X ) )
25 23 24 eqtr4d 
 |-  ( ( ( N e. NN0 /\ X e. A ) /\ N = 0 ) -> ( N ( .g ` H ) X ) = ( N .* X ) )
26 elnn0 
 |-  ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
27 26 biimpi 
 |-  ( N e. NN0 -> ( N e. NN \/ N = 0 ) )
28 27 adantr 
 |-  ( ( N e. NN0 /\ X e. A ) -> ( N e. NN \/ N = 0 ) )
29 9 25 28 mpjaodan 
 |-  ( ( N e. NN0 /\ X e. A ) -> ( N ( .g ` H ) X ) = ( N .* X ) )