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 ⊢ + = ( +g ‘ 𝑈 )
lcfrlem17.z ⊢ 0 = ( 0g ‘ 𝑈 )
lcfrlem17.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
lcfrlem17.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
lcfrlem17.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
lcfrlem17.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
lcfrlem17.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
lcfrlem17.ne ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
lcfrlem22.b ⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
lcfrlem24.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
lcfrlem24.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
lcfrlem24.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
lcfrlem24.r ⊢ 𝑅 = ( Base ‘ 𝑆 )
lcfrlem24.j ⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
lcfrlem24.ib ⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
lcfrlem24.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
lcfrlem25.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
lcfrlem28.jn ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
lcfrlem29.i ⊢ 𝐹 = ( invr ‘ 𝑆 )
lcfrlem30.m ⊢ − = ( -g ‘ 𝐷 )
lcfrlem30.c ⊢ 𝐶 = ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
Assertion lcfrlem36 ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )

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 ⊢ + = ( +g ‘ 𝑈 )
6 lcfrlem17.z ⊢ 0 = ( 0g ‘ 𝑈 )
7 lcfrlem17.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8 lcfrlem17.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
9 lcfrlem17.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 lcfrlem17.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11 lcfrlem17.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
12 lcfrlem17.ne ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
13 lcfrlem22.b ⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
14 lcfrlem24.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
15 lcfrlem24.s ⊢ 𝑆 = ( Scalar ‘ 𝑈 )
16 lcfrlem24.q ⊢ 𝑄 = ( 0g ‘ 𝑆 )
17 lcfrlem24.r ⊢ 𝑅 = ( Base ‘ 𝑆 )
18 lcfrlem24.j ⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
19 lcfrlem24.ib ⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
20 lcfrlem24.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
21 lcfrlem25.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
22 lcfrlem28.jn ⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
23 lcfrlem29.i ⊢ 𝐹 = ( invr ‘ 𝑆 )
24 lcfrlem30.m ⊢ − = ( -g ‘ 𝐷 )
25 lcfrlem30.c ⊢ 𝐶 = ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
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 ) } ) = ( 𝐿 ‘ 𝐶 ) )
30 29 fveq2d ⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )
31 28 30 eqtr3d ⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) = ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )
32 eqimss ⊢ ( ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) = ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )
33 31 32 syl ⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )
34 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
35 1 3 9 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ 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 ⊢ ( 𝜑 → 𝐶 ∈ ( LFnl ‘ 𝑈 ) )
38 4 36 20 35 37 lkrssv ⊢ ( 𝜑 → ( 𝐿 ‘ 𝐶 ) ⊆ 𝑉 )
39 1 3 4 34 2 dochlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐿 ‘ 𝐶 ) ⊆ 𝑉 ) → ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
40 9 38 39 syl2anc ⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
41 4 34 7 35 40 27 lspsnel5 ⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) ↔ ( 𝑁 ‘ { ( 𝑋 + 𝑌 ) } ) ⊆ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) ) )
42 33 41 mpbird ⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐶 ) ) )