Step |
Hyp |
Ref |
Expression |
1 |
|
hdmap1l6.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmap1l6.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmap1l6.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmap1l6.p |
⊢ + = ( +g ‘ 𝑈 ) |
5 |
|
hdmap1l6.s |
⊢ − = ( -g ‘ 𝑈 ) |
6 |
|
hdmap1l6c.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
7 |
|
hdmap1l6.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
8 |
|
hdmap1l6.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
hdmap1l6.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
10 |
|
hdmap1l6.a |
⊢ ✚ = ( +g ‘ 𝐶 ) |
11 |
|
hdmap1l6.r |
⊢ 𝑅 = ( -g ‘ 𝐶 ) |
12 |
|
hdmap1l6.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
13 |
|
hdmap1l6.l |
⊢ 𝐿 = ( LSpan ‘ 𝐶 ) |
14 |
|
hdmap1l6.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
hdmap1l6.i |
⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
hdmap1l6.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
|
hdmap1l6.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
18 |
|
hdmap1l6cl.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
19 |
|
hdmap1l6.mn |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) ) |
20 |
|
hdmap1l6i.xn |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |
21 |
|
hdmap1l6i.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
22 |
|
hdmap1l6i.z |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) ) |
23 |
|
hdmap1l6i.yz |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) |
24 |
18
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
25 |
21
|
eldifad |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
26 |
1 2 3 7 16 24 25
|
dvh3dim |
⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑉 ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
27 |
16
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
28 |
17
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝐹 ∈ 𝐷 ) |
29 |
18
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
30 |
19
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) ) |
31 |
20
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |
32 |
23
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) ) |
33 |
21
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
34 |
22
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) ) |
35 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
36 |
1 2 16
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
37 |
36
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑈 ∈ LMod ) |
38 |
3 35 7 36 24 25
|
lspprcl |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
39 |
38
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
40 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑤 ∈ 𝑉 ) |
41 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
42 |
6 35 37 39 40 41
|
lssneln0 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) ) |
43 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 27 28 29 30 31 32 33 34 42 41
|
hdmap1l6h |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) 〉 ) = ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) ) ) |
44 |
43
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑤 ∈ 𝑉 ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) 〉 ) = ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) ) ) ) |
45 |
26 44
|
mpd |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) 〉 ) = ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) ) ) |