Step |
Hyp |
Ref |
Expression |
1 |
|
hdmap14lem1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmap14lem1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmap14lem1.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmap14lem1.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
5 |
|
hdmap14lem3.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
6 |
|
hdmap14lem1.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
7 |
|
hdmap14lem1.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
8 |
|
hdmap14lem1.z |
⊢ 𝑍 = ( 0g ‘ 𝑅 ) |
9 |
|
hdmap14lem1.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
hdmap14lem2.e |
⊢ ∙ = ( ·𝑠 ‘ 𝐶 ) |
11 |
|
hdmap14lem1.l |
⊢ 𝐿 = ( LSpan ‘ 𝐶 ) |
12 |
|
hdmap14lem2.p |
⊢ 𝑃 = ( Scalar ‘ 𝐶 ) |
13 |
|
hdmap14lem2.a |
⊢ 𝐴 = ( Base ‘ 𝑃 ) |
14 |
|
hdmap14lem2.q |
⊢ 𝑄 = ( 0g ‘ 𝑃 ) |
15 |
|
hdmap14lem1.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
hdmap14lem1.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
|
hdmap14lem3.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
18 |
|
hdmap14lem6.f |
⊢ ( 𝜑 → 𝐹 = 𝑍 ) |
19 |
1 9 16
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
20 |
12 13 14
|
lmod0cl |
⊢ ( 𝐶 ∈ LMod → 𝑄 ∈ 𝐴 ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
22 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
23 |
17
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
24 |
1 2 3 9 22 15 16 23
|
hdmapcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ) |
25 |
|
eqid |
⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 ) |
26 |
22 12 10 14 25
|
lmod0vs |
⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ) → ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) ) |
27 |
19 24 26
|
syl2anc |
⊢ ( 𝜑 → ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) ) |
28 |
27
|
eqcomd |
⊢ ( 𝜑 → ( 0g ‘ 𝐶 ) = ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
29 |
|
oveq1 |
⊢ ( 𝑔 = 𝑄 → ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
30 |
29
|
rspceeqv |
⊢ ( ( 𝑄 ∈ 𝐴 ∧ ( 0g ‘ 𝐶 ) = ( 𝑄 ∙ ( 𝑆 ‘ 𝑋 ) ) ) → ∃ 𝑔 ∈ 𝐴 ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
31 |
21 28 30
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐴 ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
32 |
1 2 3 5 9 25 22 15 16 17
|
hdmapnzcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( ( Base ‘ 𝐶 ) ∖ { ( 0g ‘ 𝐶 ) } ) ) |
33 |
|
eldifsni |
⊢ ( ( 𝑆 ‘ 𝑋 ) ∈ ( ( Base ‘ 𝐶 ) ∖ { ( 0g ‘ 𝐶 ) } ) → ( 𝑆 ‘ 𝑋 ) ≠ ( 0g ‘ 𝐶 ) ) |
34 |
32 33
|
syl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ≠ ( 0g ‘ 𝐶 ) ) |
35 |
34
|
neneqd |
⊢ ( 𝜑 → ¬ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ) |
36 |
35
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ¬ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ) |
37 |
|
simp3l |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
38 |
37
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) ) |
39 |
1 9 16
|
lcdlvec |
⊢ ( 𝜑 → 𝐶 ∈ LVec ) |
40 |
39
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → 𝐶 ∈ LVec ) |
41 |
|
simp2l |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → 𝑔 ∈ 𝐴 ) |
42 |
24
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ) |
43 |
22 10 12 13 14 25 40 41 42
|
lvecvs0or |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) ↔ ( 𝑔 = 𝑄 ∨ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ) ) ) |
44 |
38 43
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( 𝑔 = 𝑄 ∨ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ) ) |
45 |
44
|
orcomd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ∨ 𝑔 = 𝑄 ) ) |
46 |
45
|
ord |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ¬ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) → 𝑔 = 𝑄 ) ) |
47 |
36 46
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → 𝑔 = 𝑄 ) |
48 |
|
simp3r |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
49 |
48
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) ) |
50 |
|
simp2r |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ℎ ∈ 𝐴 ) |
51 |
22 10 12 13 14 25 40 50 42
|
lvecvs0or |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) = ( 0g ‘ 𝐶 ) ↔ ( ℎ = 𝑄 ∨ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ) ) ) |
52 |
49 51
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ℎ = 𝑄 ∨ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ) ) |
53 |
52
|
orcomd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) ∨ ℎ = 𝑄 ) ) |
54 |
53
|
ord |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ( ¬ ( 𝑆 ‘ 𝑋 ) = ( 0g ‘ 𝐶 ) → ℎ = 𝑄 ) ) |
55 |
36 54
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → ℎ = 𝑄 ) |
56 |
47 55
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) ∧ ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) → 𝑔 = ℎ ) |
57 |
56
|
3exp |
⊢ ( 𝜑 → ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴 ) → ( ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) → 𝑔 = ℎ ) ) ) |
58 |
57
|
ralrimivv |
⊢ ( 𝜑 → ∀ 𝑔 ∈ 𝐴 ∀ ℎ ∈ 𝐴 ( ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) → 𝑔 = ℎ ) ) |
59 |
|
oveq1 |
⊢ ( 𝑔 = ℎ → ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
60 |
59
|
eqeq2d |
⊢ ( 𝑔 = ℎ → ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) |
61 |
60
|
reu4 |
⊢ ( ∃! 𝑔 ∈ 𝐴 ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ( ∃ 𝑔 ∈ 𝐴 ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ∀ 𝑔 ∈ 𝐴 ∀ ℎ ∈ 𝐴 ( ( ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ∧ ( 0g ‘ 𝐶 ) = ( ℎ ∙ ( 𝑆 ‘ 𝑋 ) ) ) → 𝑔 = ℎ ) ) ) |
62 |
31 58 61
|
sylanbrc |
⊢ ( 𝜑 → ∃! 𝑔 ∈ 𝐴 ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
63 |
18
|
oveq1d |
⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) = ( 𝑍 · 𝑋 ) ) |
64 |
1 2 16
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
65 |
3 6 4 8 5
|
lmod0vs |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑍 · 𝑋 ) = 0 ) |
66 |
64 23 65
|
syl2anc |
⊢ ( 𝜑 → ( 𝑍 · 𝑋 ) = 0 ) |
67 |
63 66
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) = 0 ) |
68 |
67
|
fveq2d |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑆 ‘ 0 ) ) |
69 |
1 2 5 9 25 15 16
|
hdmapval0 |
⊢ ( 𝜑 → ( 𝑆 ‘ 0 ) = ( 0g ‘ 𝐶 ) ) |
70 |
68 69
|
eqtrd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 0g ‘ 𝐶 ) ) |
71 |
70
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) |
72 |
71
|
reubidv |
⊢ ( 𝜑 → ( ∃! 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∃! 𝑔 ∈ 𝐴 ( 0g ‘ 𝐶 ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) ) |
73 |
62 72
|
mpbird |
⊢ ( 𝜑 → ∃! 𝑔 ∈ 𝐴 ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝑔 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |