Metamath Proof Explorer


Theorem lcfrlem14

Description: Lemma for lcfr . (Contributed by NM, 10-Mar-2015)

Ref Expression
Hypotheses lcf1o.h 𝐻 = ( LHyp ‘ 𝐾 )
lcf1o.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lcf1o.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lcf1o.v 𝑉 = ( Base ‘ 𝑈 )
lcf1o.a + = ( +g𝑈 )
lcf1o.t · = ( ·𝑠𝑈 )
lcf1o.s 𝑆 = ( Scalar ‘ 𝑈 )
lcf1o.r 𝑅 = ( Base ‘ 𝑆 )
lcf1o.z 0 = ( 0g𝑈 )
lcf1o.f 𝐹 = ( LFnl ‘ 𝑈 )
lcf1o.l 𝐿 = ( LKer ‘ 𝑈 )
lcf1o.d 𝐷 = ( LDual ‘ 𝑈 )
lcf1o.q 𝑄 = ( 0g𝐷 )
lcf1o.c 𝐶 = { 𝑓𝐹 ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) }
lcf1o.j 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣𝑉 ↦ ( 𝑘𝑅𝑤 ∈ ( ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
lcflo.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
lcfrlem10.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
lcfrlem14.n 𝑁 = ( LSpan ‘ 𝑈 )
Assertion lcfrlem14 ( 𝜑 → ( ‘ ( 𝐿 ‘ ( 𝐽𝑋 ) ) ) = ( 𝑁 ‘ { 𝑋 } ) )

Proof

Step Hyp Ref Expression
1 lcf1o.h 𝐻 = ( LHyp ‘ 𝐾 )
2 lcf1o.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 lcf1o.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 lcf1o.v 𝑉 = ( Base ‘ 𝑈 )
5 lcf1o.a + = ( +g𝑈 )
6 lcf1o.t · = ( ·𝑠𝑈 )
7 lcf1o.s 𝑆 = ( Scalar ‘ 𝑈 )
8 lcf1o.r 𝑅 = ( Base ‘ 𝑆 )
9 lcf1o.z 0 = ( 0g𝑈 )
10 lcf1o.f 𝐹 = ( LFnl ‘ 𝑈 )
11 lcf1o.l 𝐿 = ( LKer ‘ 𝑈 )
12 lcf1o.d 𝐷 = ( LDual ‘ 𝑈 )
13 lcf1o.q 𝑄 = ( 0g𝐷 )
14 lcf1o.c 𝐶 = { 𝑓𝐹 ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) }
15 lcf1o.j 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣𝑉 ↦ ( 𝑘𝑅𝑤 ∈ ( ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
16 lcflo.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 lcfrlem10.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18 lcfrlem14.n 𝑁 = ( LSpan ‘ 𝑈 )
19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 lcfrlem11 ( 𝜑 → ( 𝐿 ‘ ( 𝐽𝑋 ) ) = ( ‘ { 𝑋 } ) )
20 17 eldifad ( 𝜑𝑋𝑉 )
21 20 snssd ( 𝜑 → { 𝑋 } ⊆ 𝑉 )
22 1 3 2 4 18 16 21 dochocsp ( 𝜑 → ( ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( ‘ { 𝑋 } ) )
23 19 22 eqtr4d ( 𝜑 → ( 𝐿 ‘ ( 𝐽𝑋 ) ) = ( ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
24 23 fveq2d ( 𝜑 → ( ‘ ( 𝐿 ‘ ( 𝐽𝑋 ) ) ) = ( ‘ ( ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) )
25 eqid ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
26 1 3 4 18 25 dihlsprn ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
27 16 20 26 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
28 1 25 2 dochoc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ‘ ( ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ { 𝑋 } ) )
29 16 27 28 syl2anc ( 𝜑 → ( ‘ ( ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ { 𝑋 } ) )
30 24 29 eqtrd ( 𝜑 → ( ‘ ( 𝐿 ‘ ( 𝐽𝑋 ) ) ) = ( 𝑁 ‘ { 𝑋 } ) )