Step |
Hyp |
Ref |
Expression |
1 |
|
hgmaprnlem1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hgmaprnlem1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hgmaprnlem1.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hgmaprnlem1.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
5 |
|
hgmaprnlem1.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
6 |
|
hgmaprnlem1.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
7 |
|
hgmaprnlem1.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
8 |
|
hgmaprnlem1.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
hgmaprnlem1.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
10 |
|
hgmaprnlem1.p |
⊢ 𝑃 = ( Scalar ‘ 𝐶 ) |
11 |
|
hgmaprnlem1.a |
⊢ 𝐴 = ( Base ‘ 𝑃 ) |
12 |
|
hgmaprnlem1.e |
⊢ ∙ = ( ·𝑠 ‘ 𝐶 ) |
13 |
|
hgmaprnlem1.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
14 |
|
hgmaprnlem1.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
hgmaprnlem1.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
hgmaprnlem1.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
|
hgmaprnlem1.z |
⊢ ( 𝜑 → 𝑧 ∈ 𝐴 ) |
18 |
|
hgmaprnlem1.t2 |
⊢ ( 𝜑 → 𝑡 ∈ ( 𝑉 ∖ { 0 } ) ) |
19 |
|
hgmaprnlem1.s2 |
⊢ ( 𝜑 → 𝑠 ∈ 𝑉 ) |
20 |
|
hgmaprnlem1.sz |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑠 ) = ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) ) |
21 |
|
hgmaprnlem1.k2 |
⊢ ( 𝜑 → 𝑘 ∈ 𝐵 ) |
22 |
|
hgmaprnlem1.sk |
⊢ ( 𝜑 → 𝑠 = ( 𝑘 · 𝑡 ) ) |
23 |
22
|
fveq2d |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑠 ) = ( 𝑆 ‘ ( 𝑘 · 𝑡 ) ) ) |
24 |
18
|
eldifad |
⊢ ( 𝜑 → 𝑡 ∈ 𝑉 ) |
25 |
1 2 3 6 4 5 8 12 14 15 16 24 21
|
hgmapvs |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑘 · 𝑡 ) ) = ( ( 𝐺 ‘ 𝑘 ) ∙ ( 𝑆 ‘ 𝑡 ) ) ) |
26 |
23 20 25
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) = ( ( 𝐺 ‘ 𝑘 ) ∙ ( 𝑆 ‘ 𝑡 ) ) ) |
27 |
1 8 16
|
lcdlvec |
⊢ ( 𝜑 → 𝐶 ∈ LVec ) |
28 |
1 2 4 5 8 10 11 15 16 21
|
hgmapdcl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑘 ) ∈ 𝐴 ) |
29 |
1 2 3 8 9 14 16 24
|
hdmapcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) |
30 |
|
eldifsni |
⊢ ( 𝑡 ∈ ( 𝑉 ∖ { 0 } ) → 𝑡 ≠ 0 ) |
31 |
18 30
|
syl |
⊢ ( 𝜑 → 𝑡 ≠ 0 ) |
32 |
1 2 3 7 8 13 14 16 24
|
hdmapeq0 |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑡 ) = 𝑄 ↔ 𝑡 = 0 ) ) |
33 |
32
|
necon3bid |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑡 ) ≠ 𝑄 ↔ 𝑡 ≠ 0 ) ) |
34 |
31 33
|
mpbird |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ≠ 𝑄 ) |
35 |
9 12 10 11 13 27 17 28 29 34
|
lvecvscan2 |
⊢ ( 𝜑 → ( ( 𝑧 ∙ ( 𝑆 ‘ 𝑡 ) ) = ( ( 𝐺 ‘ 𝑘 ) ∙ ( 𝑆 ‘ 𝑡 ) ) ↔ 𝑧 = ( 𝐺 ‘ 𝑘 ) ) ) |
36 |
26 35
|
mpbid |
⊢ ( 𝜑 → 𝑧 = ( 𝐺 ‘ 𝑘 ) ) |
37 |
1 2 4 5 15 16
|
hgmapfnN |
⊢ ( 𝜑 → 𝐺 Fn 𝐵 ) |
38 |
|
fnfvelrn |
⊢ ( ( 𝐺 Fn 𝐵 ∧ 𝑘 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑘 ) ∈ ran 𝐺 ) |
39 |
37 21 38
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑘 ) ∈ ran 𝐺 ) |
40 |
36 39
|
eqeltrd |
⊢ ( 𝜑 → 𝑧 ∈ ran 𝐺 ) |