Step |
Hyp |
Ref |
Expression |
1 |
|
lcfrlem17.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcfrlem17.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcfrlem17.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
lcfrlem17.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
lcfrlem17.p |
⊢ + = ( +g ‘ 𝑈 ) |
6 |
|
lcfrlem17.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
7 |
|
lcfrlem17.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
8 |
|
lcfrlem17.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑈 ) |
9 |
|
lcfrlem17.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
lcfrlem17.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
11 |
|
lcfrlem17.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
12 |
|
lcfrlem17.ne |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
13 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
15 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
16 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) |
18 |
1 2 3 4 5 6 7 8 13 14 15 16 17
|
lcfrlem20 |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 ) |
19 |
1 3 9
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
20 |
10
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
21 |
11
|
eldifad |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
22 |
4 5
|
lmodcom |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) |
23 |
19 20 21 22
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) |
24 |
23
|
sneqd |
⊢ ( 𝜑 → { ( 𝑋 + 𝑌 ) } = { ( 𝑌 + 𝑋 ) } ) |
25 |
24
|
fveq2d |
⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) |
26 |
25
|
eleq2d |
⊢ ( 𝜑 → ( 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ↔ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) ) |
27 |
26
|
biimprd |
⊢ ( 𝜑 → ( 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) → 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
28 |
27
|
con3dimp |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) |
29 |
|
prcom |
⊢ { 𝑋 , 𝑌 } = { 𝑌 , 𝑋 } |
30 |
29
|
fveq2i |
⊢ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( 𝑁 ‘ { 𝑌 , 𝑋 } ) |
31 |
30
|
a1i |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ) |
32 |
31 25
|
ineq12d |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ∩ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ∩ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) ) |
34 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
35 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
36 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
37 |
12
|
necomd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ) |
39 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) |
40 |
1 2 3 4 5 6 7 8 34 35 36 38 39
|
lcfrlem20 |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ∩ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) ∈ 𝐴 ) |
41 |
33 40
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 ) |
42 |
28 41
|
syldan |
⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 ) |
43 |
1 2 3 4 5 6 7 8 9 10 11 12
|
lcfrlem19 |
⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ) |
44 |
18 42 43
|
mpjaodan |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 ) |