Step |
Hyp |
Ref |
Expression |
1 |
|
norm3dif.1 |
⊢ 𝐴 ∈ ℋ |
2 |
|
norm3dif.2 |
⊢ 𝐵 ∈ ℋ |
3 |
|
norm3dif.3 |
⊢ 𝐶 ∈ ℋ |
4 |
1 2
|
hvsubvali |
⊢ ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) |
5 |
1 3
|
hvsubvali |
⊢ ( 𝐴 −ℎ 𝐶 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) |
6 |
3 2
|
hvsubvali |
⊢ ( 𝐶 −ℎ 𝐵 ) = ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) |
7 |
5 6
|
oveq12i |
⊢ ( ( 𝐴 −ℎ 𝐶 ) +ℎ ( 𝐶 −ℎ 𝐵 ) ) = ( ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) +ℎ ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) |
8 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
9 |
8 3
|
hvmulcli |
⊢ ( - 1 ·ℎ 𝐶 ) ∈ ℋ |
10 |
8 2
|
hvmulcli |
⊢ ( - 1 ·ℎ 𝐵 ) ∈ ℋ |
11 |
3 10
|
hvaddcli |
⊢ ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) ∈ ℋ |
12 |
1 9 11
|
hvassi |
⊢ ( ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) +ℎ ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) = ( 𝐴 +ℎ ( ( - 1 ·ℎ 𝐶 ) +ℎ ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) ) |
13 |
9 3 10
|
hvassi |
⊢ ( ( ( - 1 ·ℎ 𝐶 ) +ℎ 𝐶 ) +ℎ ( - 1 ·ℎ 𝐵 ) ) = ( ( - 1 ·ℎ 𝐶 ) +ℎ ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) |
14 |
9 3
|
hvcomi |
⊢ ( ( - 1 ·ℎ 𝐶 ) +ℎ 𝐶 ) = ( 𝐶 +ℎ ( - 1 ·ℎ 𝐶 ) ) |
15 |
3 3
|
hvsubvali |
⊢ ( 𝐶 −ℎ 𝐶 ) = ( 𝐶 +ℎ ( - 1 ·ℎ 𝐶 ) ) |
16 |
|
hvsubid |
⊢ ( 𝐶 ∈ ℋ → ( 𝐶 −ℎ 𝐶 ) = 0ℎ ) |
17 |
3 16
|
ax-mp |
⊢ ( 𝐶 −ℎ 𝐶 ) = 0ℎ |
18 |
14 15 17
|
3eqtr2i |
⊢ ( ( - 1 ·ℎ 𝐶 ) +ℎ 𝐶 ) = 0ℎ |
19 |
18
|
oveq1i |
⊢ ( ( ( - 1 ·ℎ 𝐶 ) +ℎ 𝐶 ) +ℎ ( - 1 ·ℎ 𝐵 ) ) = ( 0ℎ +ℎ ( - 1 ·ℎ 𝐵 ) ) |
20 |
|
ax-hv0cl |
⊢ 0ℎ ∈ ℋ |
21 |
20 10
|
hvcomi |
⊢ ( 0ℎ +ℎ ( - 1 ·ℎ 𝐵 ) ) = ( ( - 1 ·ℎ 𝐵 ) +ℎ 0ℎ ) |
22 |
|
ax-hvaddid |
⊢ ( ( - 1 ·ℎ 𝐵 ) ∈ ℋ → ( ( - 1 ·ℎ 𝐵 ) +ℎ 0ℎ ) = ( - 1 ·ℎ 𝐵 ) ) |
23 |
10 22
|
ax-mp |
⊢ ( ( - 1 ·ℎ 𝐵 ) +ℎ 0ℎ ) = ( - 1 ·ℎ 𝐵 ) |
24 |
19 21 23
|
3eqtri |
⊢ ( ( ( - 1 ·ℎ 𝐶 ) +ℎ 𝐶 ) +ℎ ( - 1 ·ℎ 𝐵 ) ) = ( - 1 ·ℎ 𝐵 ) |
25 |
13 24
|
eqtr3i |
⊢ ( ( - 1 ·ℎ 𝐶 ) +ℎ ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) = ( - 1 ·ℎ 𝐵 ) |
26 |
25
|
oveq2i |
⊢ ( 𝐴 +ℎ ( ( - 1 ·ℎ 𝐶 ) +ℎ ( 𝐶 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) |
27 |
7 12 26
|
3eqtri |
⊢ ( ( 𝐴 −ℎ 𝐶 ) +ℎ ( 𝐶 −ℎ 𝐵 ) ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) |
28 |
4 27
|
eqtr4i |
⊢ ( 𝐴 −ℎ 𝐵 ) = ( ( 𝐴 −ℎ 𝐶 ) +ℎ ( 𝐶 −ℎ 𝐵 ) ) |
29 |
28
|
fveq2i |
⊢ ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( normℎ ‘ ( ( 𝐴 −ℎ 𝐶 ) +ℎ ( 𝐶 −ℎ 𝐵 ) ) ) |
30 |
1 3
|
hvsubcli |
⊢ ( 𝐴 −ℎ 𝐶 ) ∈ ℋ |
31 |
3 2
|
hvsubcli |
⊢ ( 𝐶 −ℎ 𝐵 ) ∈ ℋ |
32 |
30 31
|
norm-ii-i |
⊢ ( normℎ ‘ ( ( 𝐴 −ℎ 𝐶 ) +ℎ ( 𝐶 −ℎ 𝐵 ) ) ) ≤ ( ( normℎ ‘ ( 𝐴 −ℎ 𝐶 ) ) + ( normℎ ‘ ( 𝐶 −ℎ 𝐵 ) ) ) |
33 |
29 32
|
eqbrtri |
⊢ ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) ≤ ( ( normℎ ‘ ( 𝐴 −ℎ 𝐶 ) ) + ( normℎ ‘ ( 𝐶 −ℎ 𝐵 ) ) ) |