Theorem lclkrlem2k

Description: Lemma for lclkr . Kernel closure when X is zero. (Contributed by NM, 18-Jan-2015)

	Ref	Expression
Hypotheses	lclkrlem2f.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lclkrlem2f.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lclkrlem2f.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lclkrlem2f.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lclkrlem2f.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	lclkrlem2f.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
	lclkrlem2f.z	⊢ 0 = ( 0g ‘ 𝑈 )
	lclkrlem2f.a	⊢ ⊕ = ( LSSum ‘ 𝑈 )
	lclkrlem2f.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	lclkrlem2f.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lclkrlem2f.j	⊢ 𝐽 = ( LSHyp ‘ 𝑈 )
	lclkrlem2f.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lclkrlem2f.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lclkrlem2f.p	⊢ + = ( +g ‘ 𝐷 )
	lclkrlem2f.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lclkrlem2f.b	⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) )
	lclkrlem2f.e	⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
	lclkrlem2f.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
	lclkrlem2f.le	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
	lclkrlem2f.lg	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
	lclkrlem2f.kb	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 )
	lclkrlem2f.nx	⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
	lclkrlem2k.x	⊢ ( 𝜑 → 𝑋 = 0 )
	lclkrlem2k.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	lclkrlem2k	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lclkrlem2f.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lclkrlem2f.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lclkrlem2f.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lclkrlem2f.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		lclkrlem2f.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
6		lclkrlem2f.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
7		lclkrlem2f.z	⊢ 0 = ( 0g ‘ 𝑈 )
8		lclkrlem2f.a	⊢ ⊕ = ( LSSum ‘ 𝑈 )
9		lclkrlem2f.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
10		lclkrlem2f.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
11		lclkrlem2f.j	⊢ 𝐽 = ( LSHyp ‘ 𝑈 )
12		lclkrlem2f.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
13		lclkrlem2f.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
14		lclkrlem2f.p	⊢ + = ( +g ‘ 𝐷 )
15		lclkrlem2f.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
16		lclkrlem2f.b	⊢ ( 𝜑 → 𝐵 ∈ ( 𝑉 ∖ { 0 } ) )
17		lclkrlem2f.e	⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
18		lclkrlem2f.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
19		lclkrlem2f.le	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
20		lclkrlem2f.lg	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
21		lclkrlem2f.kb	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = 𝑄 )
22		lclkrlem2f.nx	⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
23		lclkrlem2k.x	⊢ ( 𝜑 → 𝑋 = 0 )
24		lclkrlem2k.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
25	1 3 15	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
26	10 13 14 25 17 18	ldualvaddcom	⊢ ( 𝜑 → ( 𝐸 + 𝐺 ) = ( 𝐺 + 𝐸 ) )
27	26	fveq1d	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝐵 ) = ( ( 𝐺 + 𝐸 ) ‘ 𝐵 ) )
28	27 21	eqtr3d	⊢ ( 𝜑 → ( ( 𝐺 + 𝐸 ) ‘ 𝐵 ) = 𝑄 )
29	22	orcomd	⊢ ( 𝜑 → ( ¬ 𝑌 ∈ ( ⊥ ‘ { 𝐵 } ) ∨ ¬ 𝑋 ∈ ( ⊥ ‘ { 𝐵 } ) ) )
30	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 17 20 19 28 29 24 23	lclkrlem2j	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐺 + 𝐸 ) ) ) ) = ( 𝐿 ‘ ( 𝐺 + 𝐸 ) ) )
31	26	fveq2d	⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) = ( 𝐿 ‘ ( 𝐺 + 𝐸 ) ) )
32	31	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) = ( ⊥ ‘ ( 𝐿 ‘ ( 𝐺 + 𝐸 ) ) ) )
33	32	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐺 + 𝐸 ) ) ) ) )
34	30 33 31	3eqtr4d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )