Theorem lcfrlem32

Description: Lemma for lcfr . (Contributed by NM, 10-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
	lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
	lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	lcfrlem22.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
	lcfrlem24.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	lcfrlem24.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	lcfrlem24.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
	lcfrlem24.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
	lcfrlem24.j	⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
	lcfrlem24.ib	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
	lcfrlem24.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfrlem25.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lcfrlem28.jn	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
	lcfrlem29.i	⊢ 𝐹 = ( invr ‘ 𝑆 )
	lcfrlem30.m	⊢ − = ( -g ‘ 𝐷 )
	lcfrlem30.c	⊢ 𝐶 = ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
	lcfrlem31.xi	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ≠ 𝑄 )
Assertion	lcfrlem32	⊢ ( 𝜑 → 𝐶 ≠ ( 0g ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
6		lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
7		lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8		lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
9		lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11		lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
12		lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
13		lcfrlem22.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
14		lcfrlem24.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
15		lcfrlem24.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
16		lcfrlem24.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
17		lcfrlem24.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
18		lcfrlem24.j	⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
19		lcfrlem24.ib	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
20		lcfrlem24.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
21		lcfrlem25.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
22		lcfrlem28.jn	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
23		lcfrlem29.i	⊢ 𝐹 = ( invr ‘ 𝑆 )
24		lcfrlem30.m	⊢ − = ( -g ‘ 𝐷 )
25		lcfrlem30.c	⊢ 𝐶 = ( ( 𝐽 ‘ 𝑋 ) − ( ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ( ·𝑠 ‘ 𝐷 ) ( 𝐽 ‘ 𝑌 ) ) )
26		lcfrlem31.xi	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ≠ 𝑄 )
27	9	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
28	10	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
29	11	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
30	12	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
31	19	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → 𝐼 ∈ 𝐵 )
32	22	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
33	26	adantr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ≠ 𝑄 )
34		simpr	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → 𝐶 = ( 0g ‘ 𝐷 ) )
35	1 2 3 4 5 6 7 8 27 28 29 30 13 14 15 16 17 18 31 20 21 32 23 24 25 33 34	lcfrlem31	⊢ ( ( 𝜑 ∧ 𝐶 = ( 0g ‘ 𝐷 ) ) → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) )
36	35	ex	⊢ ( 𝜑 → ( 𝐶 = ( 0g ‘ 𝐷 ) → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) )
37	36	necon3d	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) → 𝐶 ≠ ( 0g ‘ 𝐷 ) ) )
38	12 37	mpd	⊢ ( 𝜑 → 𝐶 ≠ ( 0g ‘ 𝐷 ) )