Metamath Proof Explorer


Theorem lcfrlem26

Description: Lemma for lcfr . Special case of lcfrlem36 when ( ( JY )I ) is zero. (Contributed by NM, 11-Mar-2015)

Ref Expression
Hypotheses lcfrlem17.h 𝐻 = ( LHyp ‘ 𝐾 )
lcfrlem17.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lcfrlem17.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lcfrlem17.v 𝑉 = ( Base ‘ 𝑈 )
lcfrlem17.p + = ( +g𝑈 )
lcfrlem17.z 0 = ( 0g𝑈 )
lcfrlem17.n 𝑁 = ( LSpan ‘ 𝑈 )
lcfrlem17.a 𝐴 = ( LSAtoms ‘ 𝑈 )
lcfrlem17.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
lcfrlem17.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
lcfrlem17.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
lcfrlem17.ne ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
lcfrlem22.b 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ‘ { ( 𝑋 + 𝑌 ) } ) )
lcfrlem24.t · = ( ·𝑠𝑈 )
lcfrlem24.s 𝑆 = ( Scalar ‘ 𝑈 )
lcfrlem24.q 𝑄 = ( 0g𝑆 )
lcfrlem24.r 𝑅 = ( Base ‘ 𝑆 )
lcfrlem24.j 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣𝑉 ↦ ( 𝑘𝑅𝑤 ∈ ( ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
lcfrlem24.ib ( 𝜑𝐼𝐵 )
lcfrlem24.l 𝐿 = ( LKer ‘ 𝑈 )
lcfrlem25.d 𝐷 = ( LDual ‘ 𝑈 )
lcfrlem25.jz ( 𝜑 → ( ( 𝐽𝑌 ) ‘ 𝐼 ) = 𝑄 )
lcfrlem25.in ( 𝜑𝐼0 )
Assertion lcfrlem26 ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 lcfrlem17.h 𝐻 = ( LHyp ‘ 𝐾 )
2 lcfrlem17.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 lcfrlem17.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 lcfrlem17.v 𝑉 = ( Base ‘ 𝑈 )
5 lcfrlem17.p + = ( +g𝑈 )
6 lcfrlem17.z 0 = ( 0g𝑈 )
7 lcfrlem17.n 𝑁 = ( LSpan ‘ 𝑈 )
8 lcfrlem17.a 𝐴 = ( LSAtoms ‘ 𝑈 )
9 lcfrlem17.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 lcfrlem17.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11 lcfrlem17.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
12 lcfrlem17.ne ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
13 lcfrlem22.b 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ‘ { ( 𝑋 + 𝑌 ) } ) )
14 lcfrlem24.t · = ( ·𝑠𝑈 )
15 lcfrlem24.s 𝑆 = ( Scalar ‘ 𝑈 )
16 lcfrlem24.q 𝑄 = ( 0g𝑆 )
17 lcfrlem24.r 𝑅 = ( Base ‘ 𝑆 )
18 lcfrlem24.j 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣𝑉 ↦ ( 𝑘𝑅𝑤 ∈ ( ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
19 lcfrlem24.ib ( 𝜑𝐼𝐵 )
20 lcfrlem24.l 𝐿 = ( LKer ‘ 𝑈 )
21 lcfrlem25.d 𝐷 = ( LDual ‘ 𝑈 )
22 lcfrlem25.jz ( 𝜑 → ( ( 𝐽𝑌 ) ‘ 𝐼 ) = 𝑄 )
23 lcfrlem25.in ( 𝜑𝐼0 )
24 1 2 3 4 5 6 7 8 9 10 11 12 lcfrlem17 ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) )
25 24 eldifad ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑉 )
26 1 3 2 4 7 9 25 dochocsn ( 𝜑 → ( ‘ ( ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) )
27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 lcfrlem25 ( 𝜑 → ( ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿 ‘ ( 𝐽𝑌 ) ) )
28 27 fveq2d ( 𝜑 → ( ‘ ( ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )
29 26 28 eqtr3d ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) = ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )
30 eqimss ( ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) = ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )
31 29 30 syl ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )
32 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
33 1 3 9 dvhlmod ( 𝜑𝑈 ∈ LMod )
34 eqid ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
35 eqid ( 0g𝐷 ) = ( 0g𝐷 )
36 eqid { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) }
37 1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 11 lcfrlem10 ( 𝜑 → ( 𝐽𝑌 ) ∈ ( LFnl ‘ 𝑈 ) )
38 4 34 20 33 37 lkrssv ( 𝜑 → ( 𝐿 ‘ ( 𝐽𝑌 ) ) ⊆ 𝑉 )
39 1 3 4 32 2 dochlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ⊆ 𝑉 ) → ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) )
40 9 38 39 syl2anc ( 𝜑 → ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) )
41 4 32 7 33 40 25 lspsnel5 ( 𝜑 → ( ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) ↔ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) ) )
42 31 41 mpbird ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )