Step |
Hyp |
Ref |
Expression |
1 |
|
lclkrlem2m.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
2 |
|
lclkrlem2m.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
3 |
|
lclkrlem2m.s |
⊢ 𝑆 = ( Scalar ‘ 𝑈 ) |
4 |
|
lclkrlem2m.q |
⊢ × = ( .r ‘ 𝑆 ) |
5 |
|
lclkrlem2m.z |
⊢ 0 = ( 0g ‘ 𝑆 ) |
6 |
|
lclkrlem2m.i |
⊢ 𝐼 = ( invr ‘ 𝑆 ) |
7 |
|
lclkrlem2m.m |
⊢ − = ( -g ‘ 𝑈 ) |
8 |
|
lclkrlem2m.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
9 |
|
lclkrlem2m.d |
⊢ 𝐷 = ( LDual ‘ 𝑈 ) |
10 |
|
lclkrlem2m.p |
⊢ + = ( +g ‘ 𝐷 ) |
11 |
|
lclkrlem2m.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
12 |
|
lclkrlem2m.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
13 |
|
lclkrlem2m.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝐹 ) |
14 |
|
lclkrlem2m.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
15 |
|
lclkrlem2m.w |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
16 |
|
lclkrlem2m.b |
⊢ 𝐵 = ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) |
17 |
|
lclkrlem2m.n |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 ) |
18 |
|
lveclmod |
⊢ ( 𝑈 ∈ LVec → 𝑈 ∈ LMod ) |
19 |
15 18
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
20 |
|
lmodgrp |
⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Grp ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ Grp ) |
22 |
3
|
lmodring |
⊢ ( 𝑈 ∈ LMod → 𝑆 ∈ Ring ) |
23 |
19 22
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ Ring ) |
24 |
8 9 10 19 13 14
|
ldualvaddcl |
⊢ ( 𝜑 → ( 𝐸 + 𝐺 ) ∈ 𝐹 ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
26 |
3 25 1 8
|
lflcl |
⊢ ( ( 𝑈 ∈ LVec ∧ ( 𝐸 + 𝐺 ) ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ) |
27 |
15 24 11 26
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ) |
28 |
3
|
lvecdrng |
⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing ) |
29 |
15 28
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ DivRing ) |
30 |
3 25 1 8
|
lflcl |
⊢ ( ( 𝑈 ∈ LVec ∧ ( 𝐸 + 𝐺 ) ∈ 𝐹 ∧ 𝑌 ∈ 𝑉 ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) ) |
31 |
15 24 12 30
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) ) |
32 |
25 5 6
|
drnginvrcl |
⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 ) → ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ∈ ( Base ‘ 𝑆 ) ) |
33 |
29 31 17 32
|
syl3anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ∈ ( Base ‘ 𝑆 ) ) |
34 |
25 4
|
ringcl |
⊢ ( ( 𝑆 ∈ Ring ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) ) |
35 |
23 27 33 34
|
syl3anc |
⊢ ( 𝜑 → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) ) |
36 |
1 3 2 25
|
lmodvscl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) ∧ 𝑌 ∈ 𝑉 ) → ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ∈ 𝑉 ) |
37 |
19 35 12 36
|
syl3anc |
⊢ ( 𝜑 → ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ∈ 𝑉 ) |
38 |
1 7
|
grpsubcl |
⊢ ( ( 𝑈 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ∈ 𝑉 ) → ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ∈ 𝑉 ) |
39 |
21 11 37 38
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ∈ 𝑉 ) |
40 |
16 39
|
eqeltrid |
⊢ ( 𝜑 → 𝐵 ∈ 𝑉 ) |
41 |
16
|
fveq2i |
⊢ ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = ( ( 𝐸 + 𝐺 ) ‘ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ) |
42 |
|
eqid |
⊢ ( -g ‘ 𝑆 ) = ( -g ‘ 𝑆 ) |
43 |
3 42 1 7 8
|
lflsub |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐸 + 𝐺 ) ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ∈ 𝑉 ) ) → ( ( 𝐸 + 𝐺 ) ‘ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ) = ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ( -g ‘ 𝑆 ) ( ( 𝐸 + 𝐺 ) ‘ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ) ) |
44 |
19 24 11 37 43
|
syl112anc |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ) = ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ( -g ‘ 𝑆 ) ( ( 𝐸 + 𝐺 ) ‘ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ) ) |
45 |
3 25 4 1 2 8
|
lflmul |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐸 + 𝐺 ) ∈ 𝐹 ∧ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ∈ ( Base ‘ 𝑆 ) ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝐸 + 𝐺 ) ‘ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) |
46 |
19 24 35 12 45
|
syl112anc |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) |
47 |
25 4
|
ringass |
⊢ ( ( 𝑆 ∈ Ring ∧ ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) ) ) → ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) = ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ) |
48 |
23 27 33 31 47
|
syl13anc |
⊢ ( 𝜑 → ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) = ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) ) |
49 |
|
eqid |
⊢ ( 1r ‘ 𝑆 ) = ( 1r ‘ 𝑆 ) |
50 |
25 5 4 49 6
|
drnginvrl |
⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 ) → ( ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) = ( 1r ‘ 𝑆 ) ) |
51 |
29 31 17 50
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) = ( 1r ‘ 𝑆 ) ) |
52 |
51
|
oveq2d |
⊢ ( 𝜑 → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) = ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 1r ‘ 𝑆 ) ) ) |
53 |
48 52
|
eqtrd |
⊢ ( 𝜑 → ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) × ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) = ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 1r ‘ 𝑆 ) ) ) |
54 |
25 4 49
|
ringridm |
⊢ ( ( 𝑆 ∈ Ring ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 1r ‘ 𝑆 ) ) = ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ) |
55 |
23 27 54
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 1r ‘ 𝑆 ) ) = ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ) |
56 |
46 53 55
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ) |
57 |
56
|
oveq2d |
⊢ ( 𝜑 → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ( -g ‘ 𝑆 ) ( ( 𝐸 + 𝐺 ) ‘ ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ) = ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ( -g ‘ 𝑆 ) ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ) ) |
58 |
|
ringgrp |
⊢ ( 𝑆 ∈ Ring → 𝑆 ∈ Grp ) |
59 |
23 58
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ Grp ) |
60 |
25 5 42
|
grpsubid |
⊢ ( ( 𝑆 ∈ Grp ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ( -g ‘ 𝑆 ) ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ) = 0 ) |
61 |
59 27 60
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ( -g ‘ 𝑆 ) ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) ) = 0 ) |
62 |
44 57 61
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ) = 0 ) |
63 |
41 62
|
syl5eq |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 0 ) |
64 |
40 63
|
jca |
⊢ ( 𝜑 → ( 𝐵 ∈ 𝑉 ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 0 ) ) |