Step |
Hyp |
Ref |
Expression |
1 |
|
hdmap14lem8.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmap14lem8.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmap14lem8.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmap14lem8.q |
⊢ + = ( +g ‘ 𝑈 ) |
5 |
|
hdmap14lem8.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
6 |
|
hdmap14lem8.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
7 |
|
hdmap14lem8.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
8 |
|
hdmap14lem8.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
9 |
|
hdmap14lem8.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
10 |
|
hdmap14lem8.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
hdmap14lem8.d |
⊢ ✚ = ( +g ‘ 𝐶 ) |
12 |
|
hdmap14lem8.e |
⊢ ∙ = ( ·𝑠 ‘ 𝐶 ) |
13 |
|
hdmap14lem8.p |
⊢ 𝑃 = ( Scalar ‘ 𝐶 ) |
14 |
|
hdmap14lem8.a |
⊢ 𝐴 = ( Base ‘ 𝑃 ) |
15 |
|
hdmap14lem8.s |
⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
hdmap14lem8.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
|
hdmap14lem8.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
18 |
|
hdmap14lem8.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
19 |
|
hdmap14lem8.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
20 |
|
hdmap14lem8.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐴 ) |
21 |
|
hdmap14lem8.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝐴 ) |
22 |
|
hdmap14lem8.xx |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ) |
23 |
|
hdmap14lem8.yy |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) = ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) |
24 |
|
hdmap14lem8.ne |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
25 |
|
hdmap14lem8.j |
⊢ ( 𝜑 → 𝐽 ∈ 𝐴 ) |
26 |
|
hdmap14lem8.xy |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) ) |
27 |
1 10 16
|
lcdlmod |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
28 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
29 |
17
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
30 |
1 2 3 10 28 15 16 29
|
hdmapcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ) |
31 |
18
|
eldifad |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
32 |
1 2 3 10 28 15 16 31
|
hdmapcl |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ 𝐶 ) ) |
33 |
28 11 13 12 14
|
lmodvsdi |
⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝐽 ∈ 𝐴 ∧ ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) ) |
34 |
27 25 30 32 33
|
syl13anc |
⊢ ( 𝜑 → ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) ) |
35 |
1 2 3 4 10 11 15 16 29 31
|
hdmapadd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) |
36 |
35
|
oveq2d |
⊢ ( 𝜑 → ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) = ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) ) |
37 |
1 2 16
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
38 |
3 4 8 5 9
|
lmodvsdi |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐹 · ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) ) |
39 |
37 19 29 31 38
|
syl13anc |
⊢ ( 𝜑 → ( 𝐹 · ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) ) |
40 |
39
|
fveq2d |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑆 ‘ ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) ) ) |
41 |
3 8 5 9
|
lmodvscl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐹 · 𝑋 ) ∈ 𝑉 ) |
42 |
37 19 29 41
|
syl3anc |
⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) ∈ 𝑉 ) |
43 |
3 8 5 9
|
lmodvscl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉 ) → ( 𝐹 · 𝑌 ) ∈ 𝑉 ) |
44 |
37 19 31 43
|
syl3anc |
⊢ ( 𝜑 → ( 𝐹 · 𝑌 ) ∈ 𝑉 ) |
45 |
1 2 3 4 10 11 15 16 42 44
|
hdmapadd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) ) = ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ✚ ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) ) ) |
46 |
22 23
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ✚ ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) ) |
47 |
40 45 46
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) ) |
48 |
26 47
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) ) |
49 |
36 48
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) ) |
50 |
34 49
|
eqtr3d |
⊢ ( 𝜑 → ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) ) |