Metamath Proof Explorer

Theorem rrx0

Description: The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023)

Ref Expression
Hypotheses rrxsca.r 
|- H = ( RR^ ` I )
rrx0.0 
|- .0. = ( I X. { 0 } )
Assertion rrx0 
|- ( I e. V -> ( 0g ` H ) = .0. )

Proof

Step Hyp Ref Expression
1 rrxsca.r 
 |-  H = ( RR^ ` I )
2 rrx0.0 
 |-  .0. = ( I X. { 0 } )
3 1 rrxval 
 |-  ( I e. V -> H = ( toCPreHil ` ( RRfld freeLMod I ) ) )
4 3 fveq2d 
 |-  ( I e. V -> ( 0g ` H ) = ( 0g ` ( toCPreHil ` ( RRfld freeLMod I ) ) ) )
5 eqid 
 |-  ( toCPreHil ` ( RRfld freeLMod I ) ) = ( toCPreHil ` ( RRfld freeLMod I ) )
6 eqid 
 |-  ( Base ` ( RRfld freeLMod I ) ) = ( Base ` ( RRfld freeLMod I ) )
7 eqid 
 |-  ( .i ` ( RRfld freeLMod I ) ) = ( .i ` ( RRfld freeLMod I ) )
8 5 6 7 tcphval 
 |-  ( toCPreHil ` ( RRfld freeLMod I ) ) = ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) )
9 8 a1i 
 |-  ( I e. V -> ( toCPreHil ` ( RRfld freeLMod I ) ) = ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) )
10 9 fveq2d 
 |-  ( I e. V -> ( 0g ` ( toCPreHil ` ( RRfld freeLMod I ) ) ) = ( 0g ` ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) ) )
11 fvexd 
 |-  ( I e. V -> ( Base ` ( RRfld freeLMod I ) ) e. _V )
12 11 mptexd 
 |-  ( I e. V -> ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) e. _V )
13 eqid 
 |-  ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) = ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) )
14 eqid 
 |-  ( 0g ` ( RRfld freeLMod I ) ) = ( 0g ` ( RRfld freeLMod I ) )
15 13 14 tng0 
 |-  ( ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) e. _V -> ( 0g ` ( RRfld freeLMod I ) ) = ( 0g ` ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) ) )
16 12 15 syl 
 |-  ( I e. V -> ( 0g ` ( RRfld freeLMod I ) ) = ( 0g ` ( ( RRfld freeLMod I ) toNrmGrp ( x e. ( Base ` ( RRfld freeLMod I ) ) |-> ( sqrt ` ( x ( .i ` ( RRfld freeLMod I ) ) x ) ) ) ) ) )
17 refld 
 |-  RRfld e. Field
18 isfld 
 |-  ( RRfld e. Field <-> ( RRfld e. DivRing /\ RRfld e. CRing ) )
19 drngring 
 |-  ( RRfld e. DivRing -> RRfld e. Ring )
20 19 adantr 
 |-  ( ( RRfld e. DivRing /\ RRfld e. CRing ) -> RRfld e. Ring )
21 18 20 sylbi 
 |-  ( RRfld e. Field -> RRfld e. Ring )
22 17 21 ax-mp 
 |-  RRfld e. Ring
23 eqid 
 |-  ( RRfld freeLMod I ) = ( RRfld freeLMod I )
24 re0g 
 |-  0 = ( 0g ` RRfld )
25 23 24 frlm0 
 |-  ( ( RRfld e. Ring /\ I e. V ) -> ( I X. { 0 } ) = ( 0g ` ( RRfld freeLMod I ) ) )
26 22 25 mpan 
 |-  ( I e. V -> ( I X. { 0 } ) = ( 0g ` ( RRfld freeLMod I ) ) )
27 2 26 eqtr2id 
 |-  ( I e. V -> ( 0g ` ( RRfld freeLMod I ) ) = .0. )
28 10 16 27 3eqtr2d 
 |-  ( I e. V -> ( 0g ` ( toCPreHil ` ( RRfld freeLMod I ) ) ) = .0. )
29 4 28 eqtrd 
 |-  ( I e. V -> ( 0g ` H ) = .0. )