Metamath Proof Explorer


Theorem lcdlkreqN

Description: Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015) (New usage is discouraged.)

Ref Expression
Hypotheses lcdlkreq.h 𝐻 = ( LHyp ‘ 𝐾 )
lcdlkreq.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lcdlkreq.l 𝐿 = ( LKer ‘ 𝑈 )
lcdlkreq.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
lcdlkreq.o 0 = ( 0g𝐶 )
lcdlkreq.n 𝑁 = ( LSpan ‘ 𝐶 )
lcdlkreq.v 𝑉 = ( Base ‘ 𝐶 )
lcdlkreq.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
lcdlkreq.i ( 𝜑𝐼𝑉 )
lcdlkreq.g ( 𝜑𝐺 ∈ ( 𝑁 ‘ { 𝐼 } ) )
lcdlkreq.z ( 𝜑𝐺0 )
Assertion lcdlkreqN ( 𝜑 → ( 𝐿𝐺 ) = ( 𝐿𝐼 ) )

Proof

Step Hyp Ref Expression
1 lcdlkreq.h 𝐻 = ( LHyp ‘ 𝐾 )
2 lcdlkreq.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 lcdlkreq.l 𝐿 = ( LKer ‘ 𝑈 )
4 lcdlkreq.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
5 lcdlkreq.o 0 = ( 0g𝐶 )
6 lcdlkreq.n 𝑁 = ( LSpan ‘ 𝐶 )
7 lcdlkreq.v 𝑉 = ( Base ‘ 𝐶 )
8 lcdlkreq.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 lcdlkreq.i ( 𝜑𝐼𝑉 )
10 lcdlkreq.g ( 𝜑𝐺 ∈ ( 𝑁 ‘ { 𝐼 } ) )
11 lcdlkreq.z ( 𝜑𝐺0 )
12 eqid ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
13 eqid ( LDual ‘ 𝑈 ) = ( LDual ‘ 𝑈 )
14 eqid ( 0g ‘ ( LDual ‘ 𝑈 ) ) = ( 0g ‘ ( LDual ‘ 𝑈 ) )
15 eqid ( LSpan ‘ ( LDual ‘ 𝑈 ) ) = ( LSpan ‘ ( LDual ‘ 𝑈 ) )
16 1 2 8 dvhlvec ( 𝜑𝑈 ∈ LVec )
17 1 4 7 2 12 8 9 lcdvbaselfl ( 𝜑𝐼 ∈ ( LFnl ‘ 𝑈 ) )
18 9 snssd ( 𝜑 → { 𝐼 } ⊆ 𝑉 )
19 1 2 13 15 4 7 6 8 18 lcdlsp ( 𝜑 → ( 𝑁 ‘ { 𝐼 } ) = ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) )
20 10 19 eleqtrd ( 𝜑𝐺 ∈ ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) )
21 1 2 13 14 4 5 8 lcd0v2 ( 𝜑0 = ( 0g ‘ ( LDual ‘ 𝑈 ) ) )
22 11 21 neeqtrd ( 𝜑𝐺 ≠ ( 0g ‘ ( LDual ‘ 𝑈 ) ) )
23 eldifsn ( 𝐺 ∈ ( ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) ↔ ( 𝐺 ∈ ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ∧ 𝐺 ≠ ( 0g ‘ ( LDual ‘ 𝑈 ) ) ) )
24 20 22 23 sylanbrc ( 𝜑𝐺 ∈ ( ( ( LSpan ‘ ( LDual ‘ 𝑈 ) ) ‘ { 𝐼 } ) ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) )
25 12 3 13 14 15 16 17 24 lkrlspeqN ( 𝜑 → ( 𝐿𝐺 ) = ( 𝐿𝐼 ) )