Step |
Hyp |
Ref |
Expression |
1 |
|
spansnch |
|- ( A e. ~H -> ( span ` { A } ) e. CH ) |
2 |
1
|
3ad2ant1 |
|- ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) -> ( span ` { A } ) e. CH ) |
3 |
|
simp2 |
|- ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) -> B e. ~H ) |
4 |
|
eqid |
|- ( ( projh ` ( span ` { A } ) ) ` B ) = ( ( projh ` ( span ` { A } ) ) ` B ) |
5 |
|
pjeq |
|- ( ( ( span ` { A } ) e. CH /\ B e. ~H ) -> ( ( ( projh ` ( span ` { A } ) ) ` B ) = ( ( projh ` ( span ` { A } ) ) ` B ) <-> ( ( ( projh ` ( span ` { A } ) ) ` B ) e. ( span ` { A } ) /\ E. y e. ( _|_ ` ( span ` { A } ) ) B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) ) |
6 |
4 5
|
mpbii |
|- ( ( ( span ` { A } ) e. CH /\ B e. ~H ) -> ( ( ( projh ` ( span ` { A } ) ) ` B ) e. ( span ` { A } ) /\ E. y e. ( _|_ ` ( span ` { A } ) ) B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) |
7 |
6
|
simprd |
|- ( ( ( span ` { A } ) e. CH /\ B e. ~H ) -> E. y e. ( _|_ ` ( span ` { A } ) ) B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) |
8 |
2 3 7
|
syl2anc |
|- ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) -> E. y e. ( _|_ ` ( span ` { A } ) ) B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) |
9 |
|
oveq1 |
|- ( B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) -> ( B .ih A ) = ( ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) .ih A ) ) |
10 |
9
|
ad2antll |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( B .ih A ) = ( ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) .ih A ) ) |
11 |
|
pjhcl |
|- ( ( ( span ` { A } ) e. CH /\ B e. ~H ) -> ( ( projh ` ( span ` { A } ) ) ` B ) e. ~H ) |
12 |
2 3 11
|
syl2anc |
|- ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) -> ( ( projh ` ( span ` { A } ) ) ` B ) e. ~H ) |
13 |
12
|
adantr |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( ( projh ` ( span ` { A } ) ) ` B ) e. ~H ) |
14 |
|
choccl |
|- ( ( span ` { A } ) e. CH -> ( _|_ ` ( span ` { A } ) ) e. CH ) |
15 |
1 14
|
syl |
|- ( A e. ~H -> ( _|_ ` ( span ` { A } ) ) e. CH ) |
16 |
15
|
3ad2ant1 |
|- ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) -> ( _|_ ` ( span ` { A } ) ) e. CH ) |
17 |
|
chel |
|- ( ( ( _|_ ` ( span ` { A } ) ) e. CH /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> y e. ~H ) |
18 |
16 17
|
sylan |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> y e. ~H ) |
19 |
|
simpl1 |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> A e. ~H ) |
20 |
|
ax-his2 |
|- ( ( ( ( projh ` ( span ` { A } ) ) ` B ) e. ~H /\ y e. ~H /\ A e. ~H ) -> ( ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) .ih A ) = ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) + ( y .ih A ) ) ) |
21 |
13 18 19 20
|
syl3anc |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) .ih A ) = ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) + ( y .ih A ) ) ) |
22 |
|
spansnsh |
|- ( A e. ~H -> ( span ` { A } ) e. SH ) |
23 |
22
|
adantr |
|- ( ( A e. ~H /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( span ` { A } ) e. SH ) |
24 |
|
spansnid |
|- ( A e. ~H -> A e. ( span ` { A } ) ) |
25 |
24
|
adantr |
|- ( ( A e. ~H /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> A e. ( span ` { A } ) ) |
26 |
|
simpr |
|- ( ( A e. ~H /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> y e. ( _|_ ` ( span ` { A } ) ) ) |
27 |
|
shocorth |
|- ( ( span ` { A } ) e. SH -> ( ( A e. ( span ` { A } ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( A .ih y ) = 0 ) ) |
28 |
27
|
3impib |
|- ( ( ( span ` { A } ) e. SH /\ A e. ( span ` { A } ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( A .ih y ) = 0 ) |
29 |
23 25 26 28
|
syl3anc |
|- ( ( A e. ~H /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( A .ih y ) = 0 ) |
30 |
15 17
|
sylan |
|- ( ( A e. ~H /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> y e. ~H ) |
31 |
|
orthcom |
|- ( ( A e. ~H /\ y e. ~H ) -> ( ( A .ih y ) = 0 <-> ( y .ih A ) = 0 ) ) |
32 |
30 31
|
syldan |
|- ( ( A e. ~H /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( ( A .ih y ) = 0 <-> ( y .ih A ) = 0 ) ) |
33 |
29 32
|
mpbid |
|- ( ( A e. ~H /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( y .ih A ) = 0 ) |
34 |
33
|
3ad2antl1 |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( y .ih A ) = 0 ) |
35 |
34
|
oveq2d |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) + ( y .ih A ) ) = ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) + 0 ) ) |
36 |
|
hicl |
|- ( ( ( ( projh ` ( span ` { A } ) ) ` B ) e. ~H /\ A e. ~H ) -> ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) e. CC ) |
37 |
13 19 36
|
syl2anc |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) e. CC ) |
38 |
37
|
addid1d |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) + 0 ) = ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) ) |
39 |
21 35 38
|
3eqtrd |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ y e. ( _|_ ` ( span ` { A } ) ) ) -> ( ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) .ih A ) = ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) ) |
40 |
39
|
adantrr |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) .ih A ) = ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) ) |
41 |
10 40
|
eqtrd |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( B .ih A ) = ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) ) |
42 |
41
|
oveq1d |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( ( B .ih A ) / ( ( normh ` A ) ^ 2 ) ) = ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) ) |
43 |
42
|
oveq1d |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( ( ( B .ih A ) / ( ( normh ` A ) ^ 2 ) ) .h A ) = ( ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) .h A ) ) |
44 |
|
simpl1 |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> A e. ~H ) |
45 |
|
simpl3 |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> A =/= 0h ) |
46 |
|
axpjcl |
|- ( ( ( span ` { A } ) e. CH /\ B e. ~H ) -> ( ( projh ` ( span ` { A } ) ) ` B ) e. ( span ` { A } ) ) |
47 |
2 3 46
|
syl2anc |
|- ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) -> ( ( projh ` ( span ` { A } ) ) ` B ) e. ( span ` { A } ) ) |
48 |
47
|
adantr |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( ( projh ` ( span ` { A } ) ) ` B ) e. ( span ` { A } ) ) |
49 |
|
normcan |
|- ( ( A e. ~H /\ A =/= 0h /\ ( ( projh ` ( span ` { A } ) ) ` B ) e. ( span ` { A } ) ) -> ( ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) .h A ) = ( ( projh ` ( span ` { A } ) ) ` B ) ) |
50 |
44 45 48 49
|
syl3anc |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( ( ( ( ( projh ` ( span ` { A } ) ) ` B ) .ih A ) / ( ( normh ` A ) ^ 2 ) ) .h A ) = ( ( projh ` ( span ` { A } ) ) ` B ) ) |
51 |
43 50
|
eqtr2d |
|- ( ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) /\ ( y e. ( _|_ ` ( span ` { A } ) ) /\ B = ( ( ( projh ` ( span ` { A } ) ) ` B ) +h y ) ) ) -> ( ( projh ` ( span ` { A } ) ) ` B ) = ( ( ( B .ih A ) / ( ( normh ` A ) ^ 2 ) ) .h A ) ) |
52 |
8 51
|
rexlimddv |
|- ( ( A e. ~H /\ B e. ~H /\ A =/= 0h ) -> ( ( projh ` ( span ` { A } ) ) ` B ) = ( ( ( B .ih A ) / ( ( normh ` A ) ^ 2 ) ) .h A ) ) |