Metamath Proof Explorer


Theorem lcfrlem36

Description: Lemma for lcfr . (Contributed by NM, 6-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 )
lcfrlem28.jn ( 𝜑 → ( ( 𝐽𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
lcfrlem29.i 𝐹 = ( invr𝑆 )
lcfrlem30.m = ( -g𝐷 )
lcfrlem30.c 𝐶 = ( ( 𝐽𝑋 ) ( ( ( 𝐹 ‘ ( ( 𝐽𝑌 ) ‘ 𝐼 ) ) ( .r𝑆 ) ( ( 𝐽𝑋 ) ‘ 𝐼 ) ) ( ·𝑠𝐷 ) ( 𝐽𝑌 ) ) )
Assertion lcfrlem36 ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿𝐶 ) ) )

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 lcfrlem28.jn ( 𝜑 → ( ( 𝐽𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
23 lcfrlem29.i 𝐹 = ( invr𝑆 )
24 lcfrlem30.m = ( -g𝐷 )
25 lcfrlem30.c 𝐶 = ( ( 𝐽𝑋 ) ( ( ( 𝐹 ‘ ( ( 𝐽𝑌 ) ‘ 𝐼 ) ) ( .r𝑆 ) ( ( 𝐽𝑋 ) ‘ 𝐼 ) ) ( ·𝑠𝐷 ) ( 𝐽𝑌 ) ) )
26 1 2 3 4 5 6 7 8 9 10 11 12 lcfrlem17 ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) )
27 26 eldifad ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑉 )
28 1 3 2 4 7 9 27 dochocsn ( 𝜑 → ( ‘ ( ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) )
29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 lcfrlem35 ( 𝜑 → ( ‘ { ( 𝑋 + 𝑌 ) } ) = ( 𝐿𝐶 ) )
30 29 fveq2d ( 𝜑 → ( ‘ ( ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ‘ ( 𝐿𝐶 ) ) )
31 28 30 eqtr3d ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) = ( ‘ ( 𝐿𝐶 ) ) )
32 eqimss ( ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) = ( ‘ ( 𝐿𝐶 ) ) → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ‘ ( 𝐿𝐶 ) ) )
33 31 32 syl ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ‘ ( 𝐿𝐶 ) ) )
34 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
35 1 3 9 dvhlmod ( 𝜑𝑈 ∈ LMod )
36 eqid ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 lcfrlem30 ( 𝜑𝐶 ∈ ( LFnl ‘ 𝑈 ) )
38 4 36 20 35 37 lkrssv ( 𝜑 → ( 𝐿𝐶 ) ⊆ 𝑉 )
39 1 3 4 34 2 dochlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐿𝐶 ) ⊆ 𝑉 ) → ( ‘ ( 𝐿𝐶 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
40 9 38 39 syl2anc ( 𝜑 → ( ‘ ( 𝐿𝐶 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
41 4 34 7 35 40 27 lspsnel5 ( 𝜑 → ( ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿𝐶 ) ) ↔ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ‘ ( 𝐿𝐶 ) ) ) )
42 33 41 mpbird ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ‘ ( 𝐿𝐶 ) ) )