Theorem lcfrlem39

Description: Lemma for lcfr . Eliminate J . (Contributed by NM, 11-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem38.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfrlem38.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem38.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem38.p	⊢ + = ( +g ‘ 𝑈 )
	lcfrlem38.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	lcfrlem38.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfrlem38.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lcfrlem38.q	⊢ 𝑄 = ( LSubSp ‘ 𝐷 )
	lcfrlem38.c	⊢ 𝐶 = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
	lcfrlem38.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) )
	lcfrlem38.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfrlem38.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑄 )
	lcfrlem38.gs	⊢ ( 𝜑 → 𝐺 ⊆ 𝐶 )
	lcfrlem38.xe	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
	lcfrlem38.ye	⊢ ( 𝜑 → 𝑌 ∈ 𝐸 )
	lcfrlem38.z	⊢ 0 = ( 0g ‘ 𝑈 )
	lcfrlem38.x	⊢ ( 𝜑 → 𝑋 ≠ 0 )
	lcfrlem38.y	⊢ ( 𝜑 → 𝑌 ≠ 0 )
	lcfrlem38.sp	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	lcfrlem38.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	lcfrlem38.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
	lcfrlem38.i	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
	lcfrlem38.n	⊢ ( 𝜑 → 𝐼 ≠ 0 )
Assertion	lcfrlem39	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem38.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfrlem38.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfrlem38.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfrlem38.p	⊢ + = ( +g ‘ 𝑈 )
5		lcfrlem38.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		lcfrlem38.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		lcfrlem38.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
8		lcfrlem38.q	⊢ 𝑄 = ( LSubSp ‘ 𝐷 )
9		lcfrlem38.c	⊢ 𝐶 = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
10		lcfrlem38.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘ ( 𝐿 ‘ 𝑔 ) )
11		lcfrlem38.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		lcfrlem38.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑄 )
13		lcfrlem38.gs	⊢ ( 𝜑 → 𝐺 ⊆ 𝐶 )
14		lcfrlem38.xe	⊢ ( 𝜑 → 𝑋 ∈ 𝐸 )
15		lcfrlem38.ye	⊢ ( 𝜑 → 𝑌 ∈ 𝐸 )
16		lcfrlem38.z	⊢ 0 = ( 0g ‘ 𝑈 )
17		lcfrlem38.x	⊢ ( 𝜑 → 𝑋 ≠ 0 )
18		lcfrlem38.y	⊢ ( 𝜑 → 𝑌 ≠ 0 )
19		lcfrlem38.sp	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
20		lcfrlem38.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
21		lcfrlem38.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
22		lcfrlem38.i	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
23		lcfrlem38.n	⊢ ( 𝜑 → 𝐼 ≠ 0 )
24		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
25		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
26		eqid	⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
27		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
28		oveq1	⊢ ( 𝑗 = 𝑘 → ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) = ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) )
29	28	oveq2d	⊢ ( 𝑗 = 𝑘 → ( 𝑤 + ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( 𝑤 + ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) )
30	29	eqeq2d	⊢ ( 𝑗 = 𝑘 → ( 𝑣 = ( 𝑤 + ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ↔ 𝑣 = ( 𝑤 + ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) )
31	30	rexbidv	⊢ ( 𝑗 = 𝑘 → ( ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ↔ ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) )
32	31	cbvriotavw	⊢ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) = ( ℩ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) )
33	32	mpteq2i	⊢ ( 𝑣 ∈ ( Base ‘ 𝑈 ) ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) = ( 𝑣 ∈ ( Base ‘ 𝑈 ) ↦ ( ℩ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) )
34	33	mpteq2i	⊢ ( 𝑥 ∈ ( ( Base ‘ 𝑈 ) ∖ { 0 } ) ↦ ( 𝑣 ∈ ( Base ‘ 𝑈 ) ↦ ( ℩ 𝑗 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ( ( Base ‘ 𝑈 ) ∖ { 0 } ) ↦ ( 𝑣 ∈ ( Base ‘ 𝑈 ) ↦ ( ℩ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) ) ) )
35	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 26 27 34	lcfrlem38	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )