Step |
Hyp |
Ref |
Expression |
1 |
|
hdmap14lem1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmap14lem1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmap14lem1.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmap14lem1.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
5 |
|
hdmap14lem3.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
6 |
|
hdmap14lem1.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
7 |
|
hdmap14lem1.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
8 |
|
hdmap14lem1.z |
⊢ 𝑍 = ( 0g ‘ 𝑅 ) |
9 |
|
hdmap14lem1.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
hdmap14lem2.e |
⊢ ∙ = ( ·𝑠 ‘ 𝐶 ) |
11 |
|
hdmap14lem1.l |
⊢ 𝐿 = ( LSpan ‘ 𝐶 ) |
12 |
|
hdmap14lem2.p |
⊢ 𝑃 = ( Scalar ‘ 𝐶 ) |
13 |
|
hdmap14lem2.a |
⊢ 𝐴 = ( Base ‘ 𝑃 ) |
14 |
|
hdmap14lem2.q |
⊢ 𝑄 = ( 0g ‘ 𝑃 ) |
15 |
|
hdmap14lem1.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
hdmap14lem1.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
|
hdmap14lem3.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
18 |
|
hdmap14lem1.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) ) |
19 |
17
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
20 |
18
|
eldifad |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
21 |
|
eldifsni |
⊢ ( 𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) → 𝐹 ≠ 𝑍 ) |
22 |
18 21
|
syl |
⊢ ( 𝜑 → 𝐹 ≠ 𝑍 ) |
23 |
1 2 3 4 6 7 9 10 11 12 13 15 16 19 20 8 22
|
hdmap14lem1a |
⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑋 ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) ) |