| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hlhillvec.h |
|- H = ( LHyp ` K ) |
| 2 |
|
hlhillvec.u |
|- U = ( ( HLHil ` K ) ` W ) |
| 3 |
|
hlhillvec.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 4 |
|
hlhildrng.r |
|- R = ( Scalar ` U ) |
| 5 |
|
hlhilsrng.l |
|- L = ( ( DVecH ` K ) ` W ) |
| 6 |
|
hlhilsrng.s |
|- S = ( Scalar ` L ) |
| 7 |
|
hlhilsrng.b |
|- B = ( Base ` S ) |
| 8 |
|
hlhilsrng.p |
|- .+ = ( +g ` S ) |
| 9 |
|
hlhilsrng.t |
|- .x. = ( .r ` S ) |
| 10 |
|
hlhilsrng.g |
|- G = ( ( HGMap ` K ) ` W ) |
| 11 |
1 5 6 2 4 3 7
|
hlhilsbase2 |
|- ( ph -> B = ( Base ` R ) ) |
| 12 |
1 5 6 2 4 3 8
|
hlhilsplus2 |
|- ( ph -> .+ = ( +g ` R ) ) |
| 13 |
1 5 6 2 4 3 9
|
hlhilsmul2 |
|- ( ph -> .x. = ( .r ` R ) ) |
| 14 |
1 2 4 10 3
|
hlhilnvl |
|- ( ph -> G = ( *r ` R ) ) |
| 15 |
1 2 3 4
|
hlhildrng |
|- ( ph -> R e. DivRing ) |
| 16 |
|
drngring |
|- ( R e. DivRing -> R e. Ring ) |
| 17 |
15 16
|
syl |
|- ( ph -> R e. Ring ) |
| 18 |
3
|
adantr |
|- ( ( ph /\ x e. B ) -> ( K e. HL /\ W e. H ) ) |
| 19 |
|
simpr |
|- ( ( ph /\ x e. B ) -> x e. B ) |
| 20 |
1 5 6 7 10 18 19
|
hgmapcl |
|- ( ( ph /\ x e. B ) -> ( G ` x ) e. B ) |
| 21 |
3
|
3ad2ant1 |
|- ( ( ph /\ x e. B /\ y e. B ) -> ( K e. HL /\ W e. H ) ) |
| 22 |
|
simp2 |
|- ( ( ph /\ x e. B /\ y e. B ) -> x e. B ) |
| 23 |
|
simp3 |
|- ( ( ph /\ x e. B /\ y e. B ) -> y e. B ) |
| 24 |
1 5 6 7 8 10 21 22 23
|
hgmapadd |
|- ( ( ph /\ x e. B /\ y e. B ) -> ( G ` ( x .+ y ) ) = ( ( G ` x ) .+ ( G ` y ) ) ) |
| 25 |
1 5 6 7 9 10 21 22 23
|
hgmapmul |
|- ( ( ph /\ x e. B /\ y e. B ) -> ( G ` ( x .x. y ) ) = ( ( G ` y ) .x. ( G ` x ) ) ) |
| 26 |
1 5 6 7 10 18 19
|
hgmapvv |
|- ( ( ph /\ x e. B ) -> ( G ` ( G ` x ) ) = x ) |
| 27 |
11 12 13 14 17 20 24 25 26
|
issrngd |
|- ( ph -> R e. *Ring ) |