Theorem lclkrlem2q

Description: Lemma for lclkr . The sum has a closed kernel when B is nonzero. (Contributed by NM, 18-Jan-2015)

	Ref	Expression
Hypotheses	lclkrlem2m.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lclkrlem2m.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	lclkrlem2m.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	lclkrlem2m.q	⊢ × = ( .r ‘ 𝑆 )
	lclkrlem2m.z	⊢ 0 = ( 0g ‘ 𝑆 )
	lclkrlem2m.i	⊢ 𝐼 = ( invr ‘ 𝑆 )
	lclkrlem2m.m	⊢ − = ( -g ‘ 𝑈 )
	lclkrlem2m.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lclkrlem2m.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lclkrlem2m.p	⊢ + = ( +g ‘ 𝐷 )
	lclkrlem2m.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	lclkrlem2m.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	lclkrlem2m.e	⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
	lclkrlem2m.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
	lclkrlem2n.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	lclkrlem2n.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lclkrlem2o.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lclkrlem2o.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lclkrlem2o.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lclkrlem2o.a	⊢ ⊕ = ( LSSum ‘ 𝑈 )
	lclkrlem2o.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lclkrlem2q.le	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
	lclkrlem2q.lg	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
	lclkrlem2q.b	⊢ 𝐵 = ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) )
	lclkrlem2q.n	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
	lclkrlem2q.bn	⊢ ( 𝜑 → 𝐵 ≠ ( 0g ‘ 𝑈 ) )
Assertion	lclkrlem2q	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lclkrlem2m.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
2		lclkrlem2m.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
3		lclkrlem2m.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
4		lclkrlem2m.q	⊢ × = ( .r ‘ 𝑆 )
5		lclkrlem2m.z	⊢ 0 = ( 0g ‘ 𝑆 )
6		lclkrlem2m.i	⊢ 𝐼 = ( invr ‘ 𝑆 )
7		lclkrlem2m.m	⊢ − = ( -g ‘ 𝑈 )
8		lclkrlem2m.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
9		lclkrlem2m.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
10		lclkrlem2m.p	⊢ + = ( +g ‘ 𝐷 )
11		lclkrlem2m.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
12		lclkrlem2m.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
13		lclkrlem2m.e	⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
14		lclkrlem2m.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
15		lclkrlem2n.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
16		lclkrlem2n.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
17		lclkrlem2o.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
18		lclkrlem2o.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
19		lclkrlem2o.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
20		lclkrlem2o.a	⊢ ⊕ = ( LSSum ‘ 𝑈 )
21		lclkrlem2o.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22		lclkrlem2q.le	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
23		lclkrlem2q.lg	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
24		lclkrlem2q.b	⊢ 𝐵 = ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) )
25		lclkrlem2q.n	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
26		lclkrlem2q.bn	⊢ ( 𝜑 → 𝐵 ≠ ( 0g ‘ 𝑈 ) )
27		eqid	⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
28		eqid	⊢ ( LSHyp ‘ 𝑈 ) = ( LSHyp ‘ 𝑈 )
29	17 19 21	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
30	1 2 3 4 5 6 7 8 9 10 11 12 13 14 29 24 25	lclkrlem2m	⊢ ( 𝜑 → ( 𝐵 ∈ 𝑉 ∧ ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 0 ) )
31	30	simpld	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
32		eldifsn	⊢ ( 𝐵 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑈 ) } ) ↔ ( 𝐵 ∈ 𝑉 ∧ 𝐵 ≠ ( 0g ‘ 𝑈 ) ) )
33	31 26 32	sylanbrc	⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { ( 0g ‘ 𝑈 ) } ) )
34	30	simprd	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 0 )
35	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 24 25 26	lclkrlem2o	⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
36	17 18 19 1 3 5 27 20 15 8 28 16 9 10 21 33 13 14 22 23 34 35 11 12	lclkrlem2l	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )