Step |
Hyp |
Ref |
Expression |
1 |
|
spansnch |
⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Cℋ ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( span ‘ { 𝐴 } ) ∈ Cℋ ) |
3 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → 𝐵 ∈ ℋ ) |
4 |
|
eqid |
⊢ ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) = ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) |
5 |
|
pjeq |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) = ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ↔ ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ∧ ∃ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) ) ) |
6 |
4 5
|
mpbii |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ∧ ∃ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) ) |
7 |
6
|
simprd |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ∃ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) |
8 |
2 3 7
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ∃ 𝑦 ∈ ( ⊥ ‘ ( 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∧ 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) ) → ( 𝐵 ·ih 𝐴 ) = ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ·ih 𝐴 ) ) |
11 |
|
pjhcl |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ℋ ) |
12 |
2 3 11
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ℋ ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ℋ ) |
14 |
|
choccl |
⊢ ( ( span ‘ { 𝐴 } ) ∈ Cℋ → ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∈ Cℋ ) |
15 |
1 14
|
syl |
⊢ ( 𝐴 ∈ ℋ → ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∈ Cℋ ) |
16 |
15
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∈ Cℋ ) |
17 |
|
chel |
⊢ ( ( ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∈ Cℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → 𝑦 ∈ ℋ ) |
18 |
16 17
|
sylan |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → 𝑦 ∈ ℋ ) |
19 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → 𝐴 ∈ ℋ ) |
20 |
|
ax-his2 |
⊢ ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝐴 ∈ 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ·ih 𝐴 ) = ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) + ( 𝑦 ·ih 𝐴 ) ) ) |
22 |
|
spansnsh |
⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Sℋ ) |
23 |
22
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( span ‘ { 𝐴 } ) ∈ Sℋ ) |
24 |
|
spansnid |
⊢ ( 𝐴 ∈ ℋ → 𝐴 ∈ ( span ‘ { 𝐴 } ) ) |
25 |
24
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → 𝐴 ∈ ( span ‘ { 𝐴 } ) ) |
26 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) |
27 |
|
shocorth |
⊢ ( ( span ‘ { 𝐴 } ) ∈ Sℋ → ( ( 𝐴 ∈ ( span ‘ { 𝐴 } ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( 𝐴 ·ih 𝑦 ) = 0 ) ) |
28 |
27
|
3impib |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( 𝐴 ·ih 𝑦 ) = 0 ) |
29 |
23 25 26 28
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( 𝐴 ·ih 𝑦 ) = 0 ) |
30 |
15 17
|
sylan |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → 𝑦 ∈ ℋ ) |
31 |
|
orthcom |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ ) → ( ( 𝐴 ·ih 𝑦 ) = 0 ↔ ( 𝑦 ·ih 𝐴 ) = 0 ) ) |
32 |
30 31
|
syldan |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( ( 𝐴 ·ih 𝑦 ) = 0 ↔ ( 𝑦 ·ih 𝐴 ) = 0 ) ) |
33 |
29 32
|
mpbid |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( 𝑦 ·ih 𝐴 ) = 0 ) |
34 |
33
|
3ad2antl1 |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( 𝑦 ·ih 𝐴 ) = 0 ) |
35 |
34
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) + ( 𝑦 ·ih 𝐴 ) ) = ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) + 0 ) ) |
36 |
|
hicl |
⊢ ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) ∈ ℂ ) |
37 |
13 19 36
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) ∈ ℂ ) |
38 |
37
|
addid1d |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) + 0 ) = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) ) |
39 |
21 35 38
|
3eqtrd |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ) → ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ·ih 𝐴 ) = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) ) |
40 |
39
|
adantrr |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∧ 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) ) → ( 𝐵 ·ih 𝐴 ) = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) ) |
42 |
41
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∧ 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) ) → 𝐴 ∈ ℋ ) |
45 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∧ 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) ) → 𝐴 ≠ 0ℎ ) |
46 |
|
axpjcl |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) |
47 |
2 3 46
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) |
48 |
47
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( span ‘ { 𝐴 } ) ) ∧ 𝐵 = ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) +ℎ 𝑦 ) ) ) → ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) |
49 |
|
normcan |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) → ( ( ( ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ·ih 𝐴 ) / ( ( normℎ ‘ 𝐴 ) ↑ 2 ) ) ·ℎ 𝐴 ) = ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) ) |
50 |
44 45 48 49
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) ∧ ( 𝑦 ∈ ( ⊥ ‘ ( 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( projℎ ‘ ( span ‘ { 𝐴 } ) ) ‘ 𝐵 ) = ( ( ( 𝐵 ·ih 𝐴 ) / ( ( normℎ ‘ 𝐴 ) ↑ 2 ) ) ·ℎ 𝐴 ) ) |