Theorem lcfrlem41

Description: Lemma for lcfr . Eliminate span condition. (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 )
Assertion	lcfrlem41	⊢ ( 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		eqid	⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
20	11	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21	12	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝐺 ∈ 𝑄 )
22	14	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝑋 ∈ 𝐸 )
23	15	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝑌 ∈ 𝐸 )
24		simpr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) )
25	1 2 3 4 19 6 7 8 20 21 10 22 23 24	lcfrlem6	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐸 )
26	11	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
27	12	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝐺 ∈ 𝑄 )
28	13	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝐺 ⊆ 𝐶 )
29	14	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝑋 ∈ 𝐸 )
30	15	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝑌 ∈ 𝐸 )
31	17	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝑋 ≠ 0 )
32	18	adantr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → 𝑌 ≠ 0 )
33		simpr	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) )
34	1 2 3 4 5 6 7 8 9 10 26 27 28 29 30 16 31 32 19 33	lcfrlem40	⊢ ( ( 𝜑 ∧ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ≠ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑌 } ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐸 )
35	25 34	pm2.61dane	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐸 )