Metamath Proof Explorer


Theorem lcfrlem27

Description: Lemma for lcfr . Special case of lcfrlem37 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 )
lcfrlem27.g ( 𝜑𝐺 ∈ ( LSubSp ‘ 𝐷 ) )
lcfrlem27.gs ( 𝜑𝐺 ⊆ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) } )
lcfrlem27.e 𝐸 = 𝑔𝐺 ( ‘ ( 𝐿𝑔 ) )
lcfrlem27.xe ( 𝜑𝑋𝐸 )
lcfrlem27.ye ( 𝜑𝑌𝐸 )
Assertion lcfrlem27 ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )

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 lcfrlem27.g ( 𝜑𝐺 ∈ ( LSubSp ‘ 𝐷 ) )
25 lcfrlem27.gs ( 𝜑𝐺 ⊆ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) } )
26 lcfrlem27.e 𝐸 = 𝑔𝐺 ( ‘ ( 𝐿𝑔 ) )
27 lcfrlem27.xe ( 𝜑𝑋𝐸 )
28 lcfrlem27.ye ( 𝜑𝑌𝐸 )
29 eqid ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
30 eqid ( 0g𝐷 ) = ( 0g𝐷 )
31 eqid { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ‘ ( ‘ ( 𝐿𝑓 ) ) ) = ( 𝐿𝑓 ) }
32 eqid ( LSubSp ‘ 𝐷 ) = ( LSubSp ‘ 𝐷 )
33 eldifsni ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) → 𝑌0 )
34 11 33 syl ( 𝜑𝑌0 )
35 eldifsn ( 𝑌 ∈ ( 𝐸 ∖ { 0 } ) ↔ ( 𝑌𝐸𝑌0 ) )
36 28 34 35 sylanbrc ( 𝜑𝑌 ∈ ( 𝐸 ∖ { 0 } ) )
37 1 2 3 4 5 14 15 17 6 29 20 21 30 31 18 9 32 24 25 26 36 lcfrlem16 ( 𝜑 → ( 𝐽𝑌 ) ∈ 𝐺 )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 lcfrlem26 ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )
39 2fveq3 ( 𝑔 = ( 𝐽𝑌 ) → ( ‘ ( 𝐿𝑔 ) ) = ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) )
40 39 eleq2d ( 𝑔 = ( 𝐽𝑌 ) → ( ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿𝑔 ) ) ↔ ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) ) )
41 40 rspcev ( ( ( 𝐽𝑌 ) ∈ 𝐺 ∧ ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿 ‘ ( 𝐽𝑌 ) ) ) ) → ∃ 𝑔𝐺 ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿𝑔 ) ) )
42 37 38 41 syl2anc ( 𝜑 → ∃ 𝑔𝐺 ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿𝑔 ) ) )
43 eliun ( ( 𝑋 + 𝑌 ) ∈ 𝑔𝐺 ( ‘ ( 𝐿𝑔 ) ) ↔ ∃ 𝑔𝐺 ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿𝑔 ) ) )
44 42 43 sylibr ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑔𝐺 ( ‘ ( 𝐿𝑔 ) ) )
45 44 26 eleqtrrdi ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )