Theorem mapdrvallem3

Description: Lemma for mapdrval . (Contributed by NM, 2-Feb-2015)

	Ref	Expression
Hypotheses	mapdrval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdrval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapdrval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdrval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdrval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	mapdrval.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapdrval.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapdrval.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	mapdrval.t	⊢ 𝑇 = ( LSubSp ‘ 𝐷 )
	mapdrval.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
	mapdrval.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdrval.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑇 )
	mapdrval.e	⊢ ( 𝜑 → 𝑅 ⊆ 𝐶 )
	mapdrval.q	⊢ 𝑄 = ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) )
	mapdrval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdrvallem2.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	mapdrvallem2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdrvallem2.z	⊢ 0 = ( 0g ‘ 𝑈 )
	mapdrvallem2.y	⊢ 𝑌 = ( 0g ‘ 𝐷 )
Assertion	mapdrvallem3	⊢ ( 𝜑 → { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } = 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		mapdrval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdrval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdrval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
4		mapdrval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		mapdrval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
6		mapdrval.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
7		mapdrval.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
8		mapdrval.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
9		mapdrval.t	⊢ 𝑇 = ( LSubSp ‘ 𝐷 )
10		mapdrval.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
11		mapdrval.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		mapdrval.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑇 )
13		mapdrval.e	⊢ ( 𝜑 → 𝑅 ⊆ 𝐶 )
14		mapdrval.q	⊢ 𝑄 = ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) )
15		mapdrval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
16		mapdrvallem2.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
17		mapdrvallem2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
18		mapdrvallem2.z	⊢ 0 = ( 0g ‘ 𝑈 )
19		mapdrvallem2.y	⊢ 𝑌 = ( 0g ‘ 𝐷 )
20	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19	mapdrvallem2	⊢ ( 𝜑 → { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } ⊆ 𝑅 )
21		2fveq3	⊢ ( ℎ = 𝑓 → ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) = ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) )
22	21	ssiun2s	⊢ ( 𝑓 ∈ 𝑅 → ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) )
23	22	adantl	⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑅 ) → ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) )
24	23 14	sseqtrrdi	⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑅 ) → ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 )
25	13 24	ssrabdv	⊢ ( 𝜑 → 𝑅 ⊆ { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } )
26	20 25	eqssd	⊢ ( 𝜑 → { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } = 𝑅 )