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 |
|
mapdh6c.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
21 |
|
mapdh6c.z |
⊢ ( 𝜑 → 𝑍 = 0 ) |
22 |
|
mapdh6c.ne |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |
23 |
3 10 14
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
24 |
|
lmodgrp |
⊢ ( 𝐶 ∈ LMod → 𝐶 ∈ Grp ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ Grp ) |
26 |
3 5 14
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
27 |
17
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
28 |
3 5 14
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
29 |
6 8
|
lmod0vcl |
⊢ ( 𝑈 ∈ LMod → 0 ∈ 𝑉 ) |
30 |
28 29
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝑉 ) |
31 |
21 30
|
eqeltrd |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
32 |
6 9 26 27 20 31 22
|
lspindpi |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) ) |
33 |
32
|
simpld |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
34 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 33
|
mapdhcl |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ∈ 𝐷 ) |
35 |
11 19 1
|
grprid |
⊢ ( ( 𝐶 ∈ Grp ∧ ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ∈ 𝐷 ) → ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ 𝑄 ) = ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ) |
36 |
25 34 35
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ 𝑄 ) = ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ) |
37 |
21
|
oteq3d |
⊢ ( 𝜑 → 〈 𝑋 , 𝐹 , 𝑍 〉 = 〈 𝑋 , 𝐹 , 0 〉 ) |
38 |
37
|
fveq2d |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) = ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 0 〉 ) ) |
39 |
1 2 8 17 15
|
mapdhval0 |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 0 〉 ) = 𝑄 ) |
40 |
38 39
|
eqtrd |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) = 𝑄 ) |
41 |
40
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) ) = ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ 𝑄 ) ) |
42 |
21
|
oveq2d |
⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) = ( 𝑌 + 0 ) ) |
43 |
|
lmodgrp |
⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Grp ) |
44 |
28 43
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ Grp ) |
45 |
6 18 8
|
grprid |
⊢ ( ( 𝑈 ∈ Grp ∧ 𝑌 ∈ 𝑉 ) → ( 𝑌 + 0 ) = 𝑌 ) |
46 |
44 20 45
|
syl2anc |
⊢ ( 𝜑 → ( 𝑌 + 0 ) = 𝑌 ) |
47 |
42 46
|
eqtrd |
⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) = 𝑌 ) |
48 |
47
|
oteq3d |
⊢ ( 𝜑 → 〈 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) 〉 = 〈 𝑋 , 𝐹 , 𝑌 〉 ) |
49 |
48
|
fveq2d |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) 〉 ) = ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ) |
50 |
36 41 49
|
3eqtr4rd |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) 〉 ) = ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ✚ ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) ) ) |