Theorem lcfrlem21

Description: Lemma for lcfr . (Contributed by NM, 11-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 } ) )
Assertion	lcfrlem21	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 )

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	9	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14	10	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
15	11	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
16	12	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
17		simpr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
18	1 2 3 4 5 6 7 8 13 14 15 16 17	lcfrlem20	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 )
19	1 3 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
20	10	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
21	11	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
22	4 5	lmodcom	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
23	19 20 21 22	syl3anc	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
24	23	sneqd	⊢ ( 𝜑 → { ( 𝑋 + 𝑌 ) } = { ( 𝑌 + 𝑋 ) } )
25	24	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) = ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) )
26	25	eleq2d	⊢ ( 𝜑 → ( 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ↔ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) )
27	26	biimprd	⊢ ( 𝜑 → ( 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) → 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) )
28	27	con3dimp	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) )
29		prcom	⊢ { 𝑋 , 𝑌 } = { 𝑌 , 𝑋 }
30	29	fveq2i	⊢ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( 𝑁 ‘ { 𝑌 , 𝑋 } )
31	30	a1i	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( 𝑁 ‘ { 𝑌 , 𝑋 } ) )
32	31 25	ineq12d	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ∩ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) )
33	32	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) = ( ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ∩ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) )
34	9	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
35	11	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
36	10	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
37	12	necomd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
38	37	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
39		simpr	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) )
40	1 2 3 4 5 6 7 8 34 35 36 38 39	lcfrlem20	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ∩ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) ∈ 𝐴 )
41	33 40	eqeltrd	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑌 + 𝑋 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 )
42	28 41	syldan	⊢ ( ( 𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 )
43	1 2 3 4 5 6 7 8 9 10 11 12	lcfrlem19	⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ∨ ¬ 𝑌 ∈ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) )
44	18 42 43	mpjaodan	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) ) ∈ 𝐴 )