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 |
|
lclkrlem2n.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
16 |
|
lclkrlem2n.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
17 |
|
lclkrlem2o.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
18 |
|
lclkrlem2o.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
19 |
|
lclkrlem2o.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
20 |
|
lclkrlem2o.a |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
21 |
|
lclkrlem2o.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
22 |
|
lclkrlem2q.le |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) ) |
23 |
|
lclkrlem2q.lg |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) ) |
24 |
|
lclkrlem2t.n |
⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 ) |
25 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝑋 ∈ 𝑉 ) |
26 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝑌 ∈ 𝑉 ) |
27 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝐸 ∈ 𝐹 ) |
28 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝐺 ∈ 𝐹 ) |
29 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
30 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) ) |
31 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) ) |
32 |
|
eqid |
⊢ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) |
33 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 ) |
34 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) |
35 |
1 2 3 4 5 6 7 8 9 10 25 26 27 28 15 16 17 18 19 20 29 30 31 32 33 34
|
lclkrlem2s |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) |
36 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝑋 ∈ 𝑉 ) |
37 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝑌 ∈ 𝑉 ) |
38 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝐸 ∈ 𝐹 ) |
39 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝐺 ∈ 𝐹 ) |
40 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
41 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) ) |
42 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) ) |
43 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 ) |
44 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) |
45 |
1 2 3 4 5 6 7 8 9 10 36 37 38 39 15 16 17 18 19 20 40 41 42 32 43 44
|
lclkrlem2q |
⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) |
46 |
35 45
|
pm2.61dane |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) |