| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hdmap14lem8.h |
|- H = ( LHyp ` K ) |
| 2 |
|
hdmap14lem8.u |
|- U = ( ( DVecH ` K ) ` W ) |
| 3 |
|
hdmap14lem8.v |
|- V = ( Base ` U ) |
| 4 |
|
hdmap14lem8.q |
|- .+ = ( +g ` U ) |
| 5 |
|
hdmap14lem8.t |
|- .x. = ( .s ` U ) |
| 6 |
|
hdmap14lem8.o |
|- .0. = ( 0g ` U ) |
| 7 |
|
hdmap14lem8.n |
|- N = ( LSpan ` U ) |
| 8 |
|
hdmap14lem8.r |
|- R = ( Scalar ` U ) |
| 9 |
|
hdmap14lem8.b |
|- B = ( Base ` R ) |
| 10 |
|
hdmap14lem8.c |
|- C = ( ( LCDual ` K ) ` W ) |
| 11 |
|
hdmap14lem8.d |
|- .+b = ( +g ` C ) |
| 12 |
|
hdmap14lem8.e |
|- .xb = ( .s ` C ) |
| 13 |
|
hdmap14lem8.p |
|- P = ( Scalar ` C ) |
| 14 |
|
hdmap14lem8.a |
|- A = ( Base ` P ) |
| 15 |
|
hdmap14lem8.s |
|- S = ( ( HDMap ` K ) ` W ) |
| 16 |
|
hdmap14lem8.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 17 |
|
hdmap14lem8.x |
|- ( ph -> X e. ( V \ { .0. } ) ) |
| 18 |
|
hdmap14lem8.y |
|- ( ph -> Y e. ( V \ { .0. } ) ) |
| 19 |
|
hdmap14lem8.f |
|- ( ph -> F e. B ) |
| 20 |
|
hdmap14lem8.g |
|- ( ph -> G e. A ) |
| 21 |
|
hdmap14lem8.i |
|- ( ph -> I e. A ) |
| 22 |
|
hdmap14lem8.xx |
|- ( ph -> ( S ` ( F .x. X ) ) = ( G .xb ( S ` X ) ) ) |
| 23 |
|
hdmap14lem8.yy |
|- ( ph -> ( S ` ( F .x. Y ) ) = ( I .xb ( S ` Y ) ) ) |
| 24 |
|
hdmap14lem8.ne |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
| 25 |
|
hdmap14lem8.j |
|- ( ph -> J e. A ) |
| 26 |
|
hdmap14lem8.xy |
|- ( ph -> ( S ` ( F .x. ( X .+ Y ) ) ) = ( J .xb ( S ` ( X .+ Y ) ) ) ) |
| 27 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
| 28 |
|
eqid |
|- ( 0g ` C ) = ( 0g ` C ) |
| 29 |
|
eqid |
|- ( LSpan ` C ) = ( LSpan ` C ) |
| 30 |
1 10 16
|
lcdlvec |
|- ( ph -> C e. LVec ) |
| 31 |
1 2 3 6 10 28 27 15 16 17
|
hdmapnzcl |
|- ( ph -> ( S ` X ) e. ( ( Base ` C ) \ { ( 0g ` C ) } ) ) |
| 32 |
1 2 3 6 10 28 27 15 16 18
|
hdmapnzcl |
|- ( ph -> ( S ` Y ) e. ( ( Base ` C ) \ { ( 0g ` C ) } ) ) |
| 33 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
| 34 |
|
eqid |
|- ( ( mapd ` K ) ` W ) = ( ( mapd ` K ) ` W ) |
| 35 |
1 2 16
|
dvhlmod |
|- ( ph -> U e. LMod ) |
| 36 |
17
|
eldifad |
|- ( ph -> X e. V ) |
| 37 |
3 33 7
|
lspsncl |
|- ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
| 38 |
35 36 37
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
| 39 |
18
|
eldifad |
|- ( ph -> Y e. V ) |
| 40 |
3 33 7
|
lspsncl |
|- ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) ) |
| 41 |
35 39 40
|
syl2anc |
|- ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) ) |
| 42 |
1 2 33 34 16 38 41
|
mapd11 |
|- ( ph -> ( ( ( ( mapd ` K ) ` W ) ` ( N ` { X } ) ) = ( ( ( mapd ` K ) ` W ) ` ( N ` { Y } ) ) <-> ( N ` { X } ) = ( N ` { Y } ) ) ) |
| 43 |
42
|
necon3bid |
|- ( ph -> ( ( ( ( mapd ` K ) ` W ) ` ( N ` { X } ) ) =/= ( ( ( mapd ` K ) ` W ) ` ( N ` { Y } ) ) <-> ( N ` { X } ) =/= ( N ` { Y } ) ) ) |
| 44 |
24 43
|
mpbird |
|- ( ph -> ( ( ( mapd ` K ) ` W ) ` ( N ` { X } ) ) =/= ( ( ( mapd ` K ) ` W ) ` ( N ` { Y } ) ) ) |
| 45 |
1 2 3 7 10 29 34 15 16 36
|
hdmap10 |
|- ( ph -> ( ( ( mapd ` K ) ` W ) ` ( N ` { X } ) ) = ( ( LSpan ` C ) ` { ( S ` X ) } ) ) |
| 46 |
1 2 3 7 10 29 34 15 16 39
|
hdmap10 |
|- ( ph -> ( ( ( mapd ` K ) ` W ) ` ( N ` { Y } ) ) = ( ( LSpan ` C ) ` { ( S ` Y ) } ) ) |
| 47 |
44 45 46
|
3netr3d |
|- ( ph -> ( ( LSpan ` C ) ` { ( S ` X ) } ) =/= ( ( LSpan ` C ) ` { ( S ` Y ) } ) ) |
| 48 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
|
hdmap14lem8 |
|- ( ph -> ( ( J .xb ( S ` X ) ) .+b ( J .xb ( S ` Y ) ) ) = ( ( G .xb ( S ` X ) ) .+b ( I .xb ( S ` Y ) ) ) ) |
| 49 |
27 11 13 14 12 28 29 30 31 32 25 25 20 21 47 48
|
lvecindp2 |
|- ( ph -> ( J = G /\ J = I ) ) |
| 50 |
49
|
simpld |
|- ( ph -> J = G ) |
| 51 |
49
|
simprd |
|- ( ph -> J = I ) |
| 52 |
50 51
|
eqtr3d |
|- ( ph -> G = I ) |