Step |
Hyp |
Ref |
Expression |
1 |
|
lclkrlem2f.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lclkrlem2f.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lclkrlem2f.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
lclkrlem2f.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
lclkrlem2f.s |
⊢ 𝑆 = ( Scalar ‘ 𝑈 ) |
6 |
|
lclkrlem2f.q |
⊢ 𝑄 = ( 0g ‘ 𝑆 ) |
7 |
|
lclkrlem2f.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
8 |
|
lclkrlem2f.a |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
9 |
|
lclkrlem2f.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
10 |
|
lclkrlem2f.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
11 |
|
lclkrlem2f.j |
⊢ 𝐽 = ( LSHyp ‘ 𝑈 ) |
12 |
|
lclkrlem2f.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
13 |
|
lclkrlem2f.d |
⊢ 𝐷 = ( LDual ‘ 𝑈 ) |
14 |
|
lclkrlem2f.p |
⊢ + = ( +g ‘ 𝐷 ) |
15 |
|
lclkrlem2f.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
16 |
|
lclkrlem2f.b |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) ) |
17 |
|
lclkrlem2f.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝐹 ) |
18 |
|
lclkrlem2f.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
19 |
|
lclkrlem2f.le |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) ) |
20 |
|
lclkrlem2f.lg |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) ) |
21 |
|
lclkrlem2f.kb |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 ) |
22 |
|
lclkrlem2f.nx |
⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) ) |
23 |
|
lclkrlem2l.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
24 |
|
lclkrlem2l.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
25 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
26 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) ) |
27 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → 𝐸 ∈ 𝐹 ) |
28 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → 𝐺 ∈ 𝐹 ) |
29 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) ) |
30 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) ) |
31 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 ) |
32 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) ) |
33 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → 𝑋 = 0 ) |
34 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → 𝑌 ∈ 𝑉 ) |
35 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 25 26 27 28 29 30 31 32 33 34
|
lclkrlem2k |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) |
36 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
37 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) ) |
38 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝐸 ∈ 𝐹 ) |
39 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝐺 ∈ 𝐹 ) |
40 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) ) |
41 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) ) |
42 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 ) |
43 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) ) |
44 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ 𝑉 ) |
45 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 ) |
46 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 36 37 38 39 40 41 42 43 44 45
|
lclkrlem2j |
⊢ ( ( 𝜑 ∧ 𝑌 = 0 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) |
47 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
48 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) ) |
49 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝐸 ∈ 𝐹 ) |
50 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝐺 ∈ 𝐹 ) |
51 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) ) |
52 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) ) |
53 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 ) |
54 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) ) |
55 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝑋 ∈ 𝑉 ) |
56 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝑋 ≠ 0 ) |
57 |
|
eldifsn |
⊢ ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) ) |
58 |
55 56 57
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
59 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝑌 ∈ 𝑉 ) |
60 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝑌 ≠ 0 ) |
61 |
|
eldifsn |
⊢ ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) ) |
62 |
59 60 61
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
63 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 47 48 49 50 51 52 53 54 58 62
|
lclkrlem2i |
⊢ ( ( 𝜑 ∧ ( 𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) |
64 |
35 46 63
|
pm2.61da2ne |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) |