Step |
Hyp |
Ref |
Expression |
1 |
|
mapdpglem.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
mapdpglem.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
mapdpglem.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
mapdpglem.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
mapdpglem.s |
⊢ − = ( -g ‘ 𝑈 ) |
6 |
|
mapdpglem.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
7 |
|
mapdpglem.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
mapdpglem.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
mapdpglem.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
10 |
|
mapdpglem.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
11 |
|
mapdpglem1.p |
⊢ ⊕ = ( LSSum ‘ 𝐶 ) |
12 |
|
mapdpglem2.j |
⊢ 𝐽 = ( LSpan ‘ 𝐶 ) |
13 |
|
mapdpglem3.f |
⊢ 𝐹 = ( Base ‘ 𝐶 ) |
14 |
|
mapdpglem3.te |
⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
15 |
|
mapdpglem3.a |
⊢ 𝐴 = ( Scalar ‘ 𝑈 ) |
16 |
|
mapdpglem3.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
17 |
|
mapdpglem3.t |
⊢ · = ( ·𝑠 ‘ 𝐶 ) |
18 |
|
mapdpglem3.r |
⊢ 𝑅 = ( -g ‘ 𝐶 ) |
19 |
|
mapdpglem3.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
20 |
|
mapdpglem3.e |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) |
21 |
|
mapdpglem4.q |
⊢ 𝑄 = ( 0g ‘ 𝑈 ) |
22 |
|
mapdpglem.ne |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
23 |
|
mapdpglem4.jt |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) |
24 |
|
mapdpglem4.z |
⊢ 0 = ( 0g ‘ 𝐴 ) |
25 |
|
mapdpglem4.g4 |
⊢ ( 𝜑 → 𝑔 ∈ 𝐵 ) |
26 |
|
mapdpglem4.z4 |
⊢ ( 𝜑 → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) |
27 |
|
mapdpglem4.t4 |
⊢ ( 𝜑 → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) |
28 |
|
mapdpglem4.xn |
⊢ ( 𝜑 → 𝑋 ≠ 𝑄 ) |
29 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
30 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑋 ∈ 𝑉 ) |
31 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑌 ∈ 𝑉 ) |
32 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
33 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝐺 ∈ 𝐹 ) |
34 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) |
35 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
36 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) |
37 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑔 ∈ 𝐵 ) |
38 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) |
39 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) |
40 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑋 ≠ 𝑄 ) |
41 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → 𝑔 = 0 ) |
42 |
1 2 3 4 5 6 7 29 30 31 11 12 13 32 15 16 17 18 33 34 21 35 36 24 37 38 39 40 41
|
mapdpglem10 |
⊢ ( ( 𝜑 ∧ 𝑔 = 0 ) → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) |
43 |
42
|
ex |
⊢ ( 𝜑 → ( 𝑔 = 0 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) ) |
44 |
43
|
necon3d |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) → 𝑔 ≠ 0 ) ) |
45 |
22 44
|
mpd |
⊢ ( 𝜑 → 𝑔 ≠ 0 ) |