Step |
Hyp |
Ref |
Expression |
1 |
|
mapdh.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
2 |
|
mapdh.i |
⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) |
3 |
|
mapdh.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
mapdh.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
mapdh.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
mapdh.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
7 |
|
mapdh.s |
⊢ − = ( -g ‘ 𝑈 ) |
8 |
|
mapdhc.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
9 |
|
mapdh.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
10 |
|
mapdh.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
mapdh.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
12 |
|
mapdh.r |
⊢ 𝑅 = ( -g ‘ 𝐶 ) |
13 |
|
mapdh.j |
⊢ 𝐽 = ( LSpan ‘ 𝐶 ) |
14 |
|
mapdh.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
|
mapdhc.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
16 |
|
mapdh.mn |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) ) |
17 |
|
mapdhcl.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
18 |
|
mapdh.p |
⊢ + = ( +g ‘ 𝑈 ) |
19 |
|
mapdh.a |
⊢ ✚ = ( +g ‘ 𝐶 ) |
20 |
|
mapdh6b0.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
21 |
|
mapdh6b0.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
22 |
|
mapdh6b0.ne |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) = { 0 } ) |
23 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
24 |
3 5 14
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
25 |
3 5 14
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
26 |
6 23 9 25 20 21
|
lspprcl |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
27 |
6 8 9 23 24 26 17
|
lspdisjb |
⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) = { 0 } ) ) |
28 |
22 27
|
mpbird |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |