Step |
Hyp |
Ref |
Expression |
1 |
|
mapdrval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
mapdrval.o |
⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
mapdrval.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
mapdrval.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
mapdrval.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑈 ) |
6 |
|
mapdrval.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
7 |
|
mapdrval.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
8 |
|
mapdrval.d |
⊢ 𝐷 = ( LDual ‘ 𝑈 ) |
9 |
|
mapdrval.t |
⊢ 𝑇 = ( LSubSp ‘ 𝐷 ) |
10 |
|
mapdrval.c |
⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) } |
11 |
|
mapdrval.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
mapdrval.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑇 ) |
13 |
|
mapdrval.e |
⊢ ( 𝜑 → 𝑅 ⊆ 𝐶 ) |
14 |
|
mapdrval.q |
⊢ 𝑄 = ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) |
15 |
|
mapdrval.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
16 |
|
mapdrvallem2.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑈 ) |
17 |
|
mapdrvallem2.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
18 |
|
mapdrvallem2.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
19 |
|
mapdrvallem2.y |
⊢ 𝑌 = ( 0g ‘ 𝐷 ) |
20 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
mapdrvallem2 |
⊢ ( 𝜑 → { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } ⊆ 𝑅 ) |
21 |
|
2fveq3 |
⊢ ( ℎ = 𝑓 → ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) = ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) |
22 |
21
|
ssiun2s |
⊢ ( 𝑓 ∈ 𝑅 → ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) ) |
23 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑅 ) → ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) ) |
24 |
23 14
|
sseqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑅 ) → ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 ) |
25 |
13 24
|
ssrabdv |
⊢ ( 𝜑 → 𝑅 ⊆ { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } ) |
26 |
20 25
|
eqssd |
⊢ ( 𝜑 → { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } = 𝑅 ) |