Step |
Hyp |
Ref |
Expression |
1 |
|
lcvfbr.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lcvfbr.c |
⊢ 𝐶 = ( ⋖L ‘ 𝑊 ) |
3 |
|
lcvfbr.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝑋 ) |
4 |
|
lcvfbr.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
5 |
|
lcvfbr.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
6 |
1 2 3 4 5
|
lcvbr |
⊢ ( 𝜑 → ( 𝑇 𝐶 𝑈 ↔ ( 𝑇 ⊊ 𝑈 ∧ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) ) ) |
7 |
|
iman |
⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ¬ ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) ∧ ¬ 𝑠 = 𝑈 ) ) |
8 |
|
anass |
⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) ∧ ¬ 𝑠 = 𝑈 ) ↔ ( 𝑇 ⊊ 𝑠 ∧ ( 𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈 ) ) ) |
9 |
|
dfpss2 |
⊢ ( 𝑠 ⊊ 𝑈 ↔ ( 𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈 ) ) |
10 |
9
|
anbi2i |
⊢ ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ↔ ( 𝑇 ⊊ 𝑠 ∧ ( 𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈 ) ) ) |
11 |
8 10
|
bitr4i |
⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) ∧ ¬ 𝑠 = 𝑈 ) ↔ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) |
12 |
7 11
|
xchbinx |
⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ¬ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) |
13 |
12
|
ralbii |
⊢ ( ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ∀ 𝑠 ∈ 𝑆 ¬ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) |
14 |
|
ralnex |
⊢ ( ∀ 𝑠 ∈ 𝑆 ¬ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ↔ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) |
15 |
13 14
|
bitri |
⊢ ( ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) |
16 |
15
|
anbi2i |
⊢ ( ( 𝑇 ⊊ 𝑈 ∧ ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ) ↔ ( 𝑇 ⊊ 𝑈 ∧ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) ) |
17 |
6 16
|
bitr4di |
⊢ ( 𝜑 → ( 𝑇 𝐶 𝑈 ↔ ( 𝑇 ⊊ 𝑈 ∧ ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ) ) ) |