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 |
|
mapdhe6.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
21 |
|
mapdhe6.z |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) ) |
22 |
|
mapdhe6.xn |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |
23 |
|
mapdh6.yz |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) |
24 |
|
mapdh6.fg |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) = 𝐺 ) |
25 |
|
mapdh6.fe |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) = 𝐸 ) |
26 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
27 |
3 5 14
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
28 |
20
|
eldifad |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
29 |
6 26 9
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
30 |
27 28 29
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
31 |
21
|
eldifad |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
32 |
6 26 9
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
33 |
27 31 32
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
34 |
|
eqid |
⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 ) |
35 |
26 34
|
lsmcl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
36 |
27 30 33 35
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
37 |
17
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
38 |
6 18
|
lmodvacl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉 ) → ( 𝑌 + 𝑍 ) ∈ 𝑉 ) |
39 |
27 28 31 38
|
syl3anc |
⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ 𝑉 ) |
40 |
6 7
|
lmodvsubcl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ ( 𝑌 + 𝑍 ) ∈ 𝑉 ) → ( 𝑋 − ( 𝑌 + 𝑍 ) ) ∈ 𝑉 ) |
41 |
27 37 39 40
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 − ( 𝑌 + 𝑍 ) ) ∈ 𝑉 ) |
42 |
6 26 9
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑋 − ( 𝑌 + 𝑍 ) ) ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
43 |
27 41 42
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
44 |
6 26 9
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
45 |
27 37 44
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
46 |
26 34
|
lsmcl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
47 |
27 43 45 46
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
48 |
3 4 5 26 14 36 47
|
mapdin |
⊢ ( 𝜑 → ( 𝑀 ‘ ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) = ( ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) ∩ ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) ) |
49 |
|
eqid |
⊢ ( LSSum ‘ 𝐶 ) = ( LSSum ‘ 𝐶 ) |
50 |
3 4 5 26 34 10 49 14 30 33
|
mapdlsm |
⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) = ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) ) ) |
51 |
3 5 14
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
52 |
6 8 9 51 28 21 37 23 22
|
lspindp2 |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) ) |
53 |
52
|
simpld |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
54 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 28 53
|
mapdhcl |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) ∈ 𝐷 ) |
55 |
24 54
|
eqeltrrd |
⊢ ( 𝜑 → 𝐺 ∈ 𝐷 ) |
56 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 55 53
|
mapdheq |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) ) |
57 |
24 56
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) |
58 |
57
|
simpld |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) |
59 |
6 8 9 51 20 31 37 23 22
|
lspindp1 |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ∧ ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) ) |
60 |
59
|
simpld |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) |
61 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 31 60
|
mapdhcl |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) ∈ 𝐷 ) |
62 |
25 61
|
eqeltrrd |
⊢ ( 𝜑 → 𝐸 ∈ 𝐷 ) |
63 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 21 62 60
|
mapdheq |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑍 〉 ) = 𝐸 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐸 ) } ) ) ) ) |
64 |
25 63
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐸 ) } ) ) ) |
65 |
64
|
simpld |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ) |
66 |
58 65
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ) |
67 |
50 66
|
eqtrd |
⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ) |
68 |
3 4 5 26 34 10 49 14 43 45
|
mapdlsm |
⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) = ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
69 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
|
mapdh6lem1N |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) ) |
70 |
69 16
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) |
71 |
68 70
|
eqtrd |
⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) = ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) |
72 |
67 71
|
ineq12d |
⊢ ( 𝜑 → ( ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) ∩ ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) = ( ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ∩ ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) ) |
73 |
48 72
|
eqtrd |
⊢ ( 𝜑 → ( 𝑀 ‘ ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) = ( ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ∩ ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) ) |
74 |
6 7 8 34 9 51 37 22 23 20 21 18
|
baerlem5b |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) = ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) |
75 |
74
|
fveq2d |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝑀 ‘ ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 − ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) ) |
76 |
3 10 14
|
lcdlvec |
⊢ ( 𝜑 → 𝐶 ∈ LVec ) |
77 |
3 4 5 6 9 10 11 13 14 15 16 37 28 55 58 31 62 65 22
|
mapdindp |
⊢ ( 𝜑 → ¬ 𝐹 ∈ ( 𝐽 ‘ { 𝐺 , 𝐸 } ) ) |
78 |
3 4 5 6 9 10 11 13 14 55 58 28 31 62 65 23
|
mapdncol |
⊢ ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ≠ ( 𝐽 ‘ { 𝐸 } ) ) |
79 |
3 4 5 6 9 10 11 13 14 55 58 8 1 20
|
mapdn0 |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐷 ∖ { 𝑄 } ) ) |
80 |
3 4 5 6 9 10 11 13 14 62 65 8 1 21
|
mapdn0 |
⊢ ( 𝜑 → 𝐸 ∈ ( 𝐷 ∖ { 𝑄 } ) ) |
81 |
11 12 1 49 13 76 15 77 78 79 80 19
|
baerlem5b |
⊢ ( 𝜑 → ( 𝐽 ‘ { ( 𝐺 ✚ 𝐸 ) } ) = ( ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ∩ ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 ✚ 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) ) |
82 |
73 75 81
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 ✚ 𝐸 ) } ) ) |