Metamath Proof Explorer

Theorem hdmap14lem8

Description: Part of proof of part 14 in Baer p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015)

Ref Expression
Hypotheses hdmap14lem8.h 
|- H = ( LHyp ` K )
hdmap14lem8.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap14lem8.v 
|- V = ( Base ` U )
hdmap14lem8.q 
|- .+ = ( +g ` U )
hdmap14lem8.t 
|- .x. = ( .s ` U )
hdmap14lem8.o 
|- .0. = ( 0g ` U )
hdmap14lem8.n 
|- N = ( LSpan ` U )
hdmap14lem8.r 
|- R = ( Scalar ` U )
hdmap14lem8.b 
|- B = ( Base ` R )
hdmap14lem8.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap14lem8.d 
|- .+b = ( +g ` C )
hdmap14lem8.e 
|- .xb = ( .s ` C )
hdmap14lem8.p 
|- P = ( Scalar ` C )
hdmap14lem8.a 
|- A = ( Base ` P )
hdmap14lem8.s 
|- S = ( ( HDMap ` K ) ` W )
hdmap14lem8.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap14lem8.x 
|- ( ph -> X e. ( V \ { .0. } ) )
hdmap14lem8.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
hdmap14lem8.f 
|- ( ph -> F e. B )
hdmap14lem8.g 
|- ( ph -> G e. A )
hdmap14lem8.i 
|- ( ph -> I e. A )
hdmap14lem8.xx 
|- ( ph -> ( S ` ( F .x. X ) ) = ( G .xb ( S ` X ) ) )
hdmap14lem8.yy 
|- ( ph -> ( S ` ( F .x. Y ) ) = ( I .xb ( S ` Y ) ) )
hdmap14lem8.ne 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
hdmap14lem8.j 
|- ( ph -> J e. A )
hdmap14lem8.xy 
|- ( ph -> ( S ` ( F .x. ( X .+ Y ) ) ) = ( J .xb ( S ` ( X .+ Y ) ) ) )
Assertion hdmap14lem8 
|- ( ph -> ( ( J .xb ( S ` X ) ) .+b ( J .xb ( S ` Y ) ) ) = ( ( G .xb ( S ` X ) ) .+b ( I .xb ( S ` Y ) ) ) )

Proof

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 1 10 16 lcdlmod 
 |-  ( ph -> C e. LMod )
28 eqid 
 |-  ( Base ` C ) = ( Base ` C )
29 17 eldifad 
 |-  ( ph -> X e. V )
30 1 2 3 10 28 15 16 29 hdmapcl 
 |-  ( ph -> ( S ` X ) e. ( Base ` C ) )
31 18 eldifad 
 |-  ( ph -> Y e. V )
32 1 2 3 10 28 15 16 31 hdmapcl 
 |-  ( ph -> ( S ` Y ) e. ( Base ` C ) )
33 28 11 13 12 14 lmodvsdi 
 |-  ( ( C e. LMod /\ ( J e. A /\ ( S ` X ) e. ( Base ` C ) /\ ( S ` Y ) e. ( Base ` C ) ) ) -> ( J .xb ( ( S ` X ) .+b ( S ` Y ) ) ) = ( ( J .xb ( S ` X ) ) .+b ( J .xb ( S ` Y ) ) ) )
34 27 25 30 32 33 syl13anc 
 |-  ( ph -> ( J .xb ( ( S ` X ) .+b ( S ` Y ) ) ) = ( ( J .xb ( S ` X ) ) .+b ( J .xb ( S ` Y ) ) ) )
35 1 2 3 4 10 11 15 16 29 31 hdmapadd 
 |-  ( ph -> ( S ` ( X .+ Y ) ) = ( ( S ` X ) .+b ( S ` Y ) ) )
36 35 oveq2d 
 |-  ( ph -> ( J .xb ( S ` ( X .+ Y ) ) ) = ( J .xb ( ( S ` X ) .+b ( S ` Y ) ) ) )
37 1 2 16 dvhlmod 
 |-  ( ph -> U e. LMod )
38 3 4 8 5 9 lmodvsdi 
 |-  ( ( U e. LMod /\ ( F e. B /\ X e. V /\ Y e. V ) ) -> ( F .x. ( X .+ Y ) ) = ( ( F .x. X ) .+ ( F .x. Y ) ) )
39 37 19 29 31 38 syl13anc 
 |-  ( ph -> ( F .x. ( X .+ Y ) ) = ( ( F .x. X ) .+ ( F .x. Y ) ) )
40 39 fveq2d 
 |-  ( ph -> ( S ` ( F .x. ( X .+ Y ) ) ) = ( S ` ( ( F .x. X ) .+ ( F .x. Y ) ) ) )
41 3 8 5 9 lmodvscl 
 |-  ( ( U e. LMod /\ F e. B /\ X e. V ) -> ( F .x. X ) e. V )
42 37 19 29 41 syl3anc 
 |-  ( ph -> ( F .x. X ) e. V )
43 3 8 5 9 lmodvscl 
 |-  ( ( U e. LMod /\ F e. B /\ Y e. V ) -> ( F .x. Y ) e. V )
44 37 19 31 43 syl3anc 
 |-  ( ph -> ( F .x. Y ) e. V )
45 1 2 3 4 10 11 15 16 42 44 hdmapadd 
 |-  ( ph -> ( S ` ( ( F .x. X ) .+ ( F .x. Y ) ) ) = ( ( S ` ( F .x. X ) ) .+b ( S ` ( F .x. Y ) ) ) )
46 22 23 oveq12d 
 |-  ( ph -> ( ( S ` ( F .x. X ) ) .+b ( S ` ( F .x. Y ) ) ) = ( ( G .xb ( S ` X ) ) .+b ( I .xb ( S ` Y ) ) ) )
47 40 45 46 3eqtrd 
 |-  ( ph -> ( S ` ( F .x. ( X .+ Y ) ) ) = ( ( G .xb ( S ` X ) ) .+b ( I .xb ( S ` Y ) ) ) )
48 26 47 eqtr3d 
 |-  ( ph -> ( J .xb ( S ` ( X .+ Y ) ) ) = ( ( G .xb ( S ` X ) ) .+b ( I .xb ( S ` Y ) ) ) )
49 36 48 eqtr3d 
 |-  ( ph -> ( J .xb ( ( S ` X ) .+b ( S ` Y ) ) ) = ( ( G .xb ( S ` X ) ) .+b ( I .xb ( S ` Y ) ) ) )
50 34 49 eqtr3d 
 |-  ( ph -> ( ( J .xb ( S ` X ) ) .+b ( J .xb ( S ` Y ) ) ) = ( ( G .xb ( S ` X ) ) .+b ( I .xb ( S ` Y ) ) ) )