Step |
Hyp |
Ref |
Expression |
1 |
|
dihval.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihval.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihval.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dihval.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
dihval.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
dihval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
dihval.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihval.d |
⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dihval.c |
⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
dihval.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
dihval.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑈 ) |
12 |
|
dihval.p |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
13 |
11
|
fvexi |
⊢ 𝑆 ∈ V |
14 |
|
nfv |
⊢ Ⅎ 𝑞 ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) |
15 |
|
nfvd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → Ⅎ 𝑞 ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
16 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dihvalc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) |
17 |
16
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) |
18 |
|
eqeq1 |
⊢ ( ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( 𝐼 ‘ 𝑋 ) → ( ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
19 |
18
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( 𝐼 ‘ 𝑋 ) ) → ( ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
20 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
21 |
|
simprl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑞 ∈ 𝐴 ) |
22 |
|
simprrl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ¬ 𝑞 ≤ 𝑊 ) |
23 |
21 22
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) |
24 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
25 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑋 ∈ 𝐵 ) |
26 |
|
simprrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) |
27 |
|
simpl3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) |
28 |
26 27
|
eqtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
29 |
1 2 3 4 5 6 8 9 10 12
|
dihjust |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
30 |
20 23 24 25 28 29
|
syl131anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
31 |
30
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
32 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dihlsscpre |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 ) |
33 |
32
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 ) |
34 |
1 2 3 4 5 6
|
lhpmcvr2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) |
35 |
34
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) |
36 |
14 15 17 19 31 33 35
|
riotasv3d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ 𝑆 ∈ V ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
37 |
13 36
|
mpan2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |