Theorem lclkrlem2t

Description: Lemma for lclkr . We eliminate all hypotheses with B here. (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 } ) )
	lclkrlem2t.n	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
Assertion	lclkrlem2t	⊢ ( 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		lclkrlem2t.n	⊢ ( 𝜑 → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
25	11	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝑋 ∈ 𝑉 )
26	12	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝑌 ∈ 𝑉 )
27	13	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝐸 ∈ 𝐹 )
28	14	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → 𝐺 ∈ 𝐹 )
29	21	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
30	22	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
31	23	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
32		eqid	⊢ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) )
33	24	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
34		simpr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) )
35	1 2 3 4 5 6 7 8 9 10 25 26 27 28 15 16 17 18 19 20 29 30 31 32 33 34	lclkrlem2s	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) = ( 0g ‘ 𝑈 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
36	11	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝑋 ∈ 𝑉 )
37	12	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝑌 ∈ 𝑉 )
38	13	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝐸 ∈ 𝐹 )
39	14	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → 𝐺 ∈ 𝐹 )
40	21	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
41	22	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐸 ) = ( ⊥ ‘ { 𝑋 } ) )
42	23	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑌 } ) )
43	24	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ≠ 0 )
44		simpr	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) )
45	1 2 3 4 5 6 7 8 9 10 36 37 38 39 15 16 17 18 19 20 40 41 42 32 43 44	lclkrlem2q	⊢ ( ( 𝜑 ∧ ( 𝑋 − ( ( ( ( 𝐸 + 𝐺 ) ‘ 𝑋 ) × ( 𝐼 ‘ ( ( 𝐸 + 𝐺 ) ‘ 𝑌 ) ) ) · 𝑌 ) ) ≠ ( 0g ‘ 𝑈 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )
46	35 45	pm2.61dane	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) ) ) = ( 𝐿 ‘ ( 𝐸 + 𝐺 ) ) )