Step |
Hyp |
Ref |
Expression |
1 |
|
ip2dii.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
ip2dii.2 |
⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) |
3 |
|
ip2dii.7 |
⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 ) |
4 |
|
ip2dii.u |
⊢ 𝑈 ∈ CPreHilOLD |
5 |
|
ip2dii.a |
⊢ 𝐴 ∈ 𝑋 |
6 |
|
ip2dii.b |
⊢ 𝐵 ∈ 𝑋 |
7 |
|
ip2dii.c |
⊢ 𝐶 ∈ 𝑋 |
8 |
|
ip2dii.d |
⊢ 𝐷 ∈ 𝑋 |
9 |
5 7 8
|
3pm3.2i |
⊢ ( 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) |
10 |
1 2 3
|
dipdi |
⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) ) → ( 𝐴 𝑃 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐴 𝑃 𝐶 ) + ( 𝐴 𝑃 𝐷 ) ) ) |
11 |
4 9 10
|
mp2an |
⊢ ( 𝐴 𝑃 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐴 𝑃 𝐶 ) + ( 𝐴 𝑃 𝐷 ) ) |
12 |
6 7 8
|
3pm3.2i |
⊢ ( 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) |
13 |
1 2 3
|
dipdi |
⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) ) → ( 𝐵 𝑃 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐵 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐷 ) ) ) |
14 |
4 12 13
|
mp2an |
⊢ ( 𝐵 𝑃 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐵 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐷 ) ) |
15 |
11 14
|
oveq12i |
⊢ ( ( 𝐴 𝑃 ( 𝐶 𝐺 𝐷 ) ) + ( 𝐵 𝑃 ( 𝐶 𝐺 𝐷 ) ) ) = ( ( ( 𝐴 𝑃 𝐶 ) + ( 𝐴 𝑃 𝐷 ) ) + ( ( 𝐵 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐷 ) ) ) |
16 |
4
|
phnvi |
⊢ 𝑈 ∈ NrmCVec |
17 |
1 2
|
nvgcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) → ( 𝐶 𝐺 𝐷 ) ∈ 𝑋 ) |
18 |
16 7 8 17
|
mp3an |
⊢ ( 𝐶 𝐺 𝐷 ) ∈ 𝑋 |
19 |
5 6 18
|
3pm3.2i |
⊢ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( 𝐶 𝐺 𝐷 ) ∈ 𝑋 ) |
20 |
1 2 3
|
dipdir |
⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( 𝐶 𝐺 𝐷 ) ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑃 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐴 𝑃 ( 𝐶 𝐺 𝐷 ) ) + ( 𝐵 𝑃 ( 𝐶 𝐺 𝐷 ) ) ) ) |
21 |
4 19 20
|
mp2an |
⊢ ( ( 𝐴 𝐺 𝐵 ) 𝑃 ( 𝐶 𝐺 𝐷 ) ) = ( ( 𝐴 𝑃 ( 𝐶 𝐺 𝐷 ) ) + ( 𝐵 𝑃 ( 𝐶 𝐺 𝐷 ) ) ) |
22 |
1 3
|
dipcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐶 ) ∈ ℂ ) |
23 |
16 5 7 22
|
mp3an |
⊢ ( 𝐴 𝑃 𝐶 ) ∈ ℂ |
24 |
1 3
|
dipcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) → ( 𝐵 𝑃 𝐷 ) ∈ ℂ ) |
25 |
16 6 8 24
|
mp3an |
⊢ ( 𝐵 𝑃 𝐷 ) ∈ ℂ |
26 |
1 3
|
dipcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐷 ) ∈ ℂ ) |
27 |
16 5 8 26
|
mp3an |
⊢ ( 𝐴 𝑃 𝐷 ) ∈ ℂ |
28 |
1 3
|
dipcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐵 𝑃 𝐶 ) ∈ ℂ ) |
29 |
16 6 7 28
|
mp3an |
⊢ ( 𝐵 𝑃 𝐶 ) ∈ ℂ |
30 |
23 25 27 29
|
add42i |
⊢ ( ( ( 𝐴 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐷 ) ) + ( ( 𝐴 𝑃 𝐷 ) + ( 𝐵 𝑃 𝐶 ) ) ) = ( ( ( 𝐴 𝑃 𝐶 ) + ( 𝐴 𝑃 𝐷 ) ) + ( ( 𝐵 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐷 ) ) ) |
31 |
15 21 30
|
3eqtr4i |
⊢ ( ( 𝐴 𝐺 𝐵 ) 𝑃 ( 𝐶 𝐺 𝐷 ) ) = ( ( ( 𝐴 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐷 ) ) + ( ( 𝐴 𝑃 𝐷 ) + ( 𝐵 𝑃 𝐶 ) ) ) |