Theorem lspsn

Description: Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014) (Proof shortened by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspsn.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	lspsn.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
	lspsn.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspsn.t	⊢ · = ( ·𝑠 ‘ 𝑊 )
	lspsn.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
Assertion	lspsn	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) = { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )

Proof

Step	Hyp	Ref	Expression
1		lspsn.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		lspsn.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3		lspsn.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
4		lspsn.t	⊢ · = ( ·𝑠 ‘ 𝑊 )
5		lspsn.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
6		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
7		simpl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑊 ∈ LMod )
8	3 1 4 2 6	lss1d	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ∈ ( LSubSp ‘ 𝑊 ) )
9		eqid	⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 )
10	1 2 9	lmod1cl	⊢ ( 𝑊 ∈ LMod → ( 1r ‘ 𝐹 ) ∈ 𝐾 )
11	3 1 4 9	lmodvs1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 1r ‘ 𝐹 ) · 𝑋 ) = 𝑋 )
12	11	eqcomd	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 = ( ( 1r ‘ 𝐹 ) · 𝑋 ) )
13		oveq1	⊢ ( 𝑘 = ( 1r ‘ 𝐹 ) → ( 𝑘 · 𝑋 ) = ( ( 1r ‘ 𝐹 ) · 𝑋 ) )
14	13	rspceeqv	⊢ ( ( ( 1r ‘ 𝐹 ) ∈ 𝐾 ∧ 𝑋 = ( ( 1r ‘ 𝐹 ) · 𝑋 ) ) → ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) )
15	10 12 14	syl2an2r	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) )
16		eqeq1	⊢ ( 𝑣 = 𝑋 → ( 𝑣 = ( 𝑘 · 𝑋 ) ↔ 𝑋 = ( 𝑘 · 𝑋 ) ) )
17	16	rexbidv	⊢ ( 𝑣 = 𝑋 → ( ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) )
18	17	elabg	⊢ ( 𝑋 ∈ 𝑉 → ( 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) )
19	18	adantl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ↔ ∃ 𝑘 ∈ 𝐾 𝑋 = ( 𝑘 · 𝑋 ) ) )
20	15 19	mpbird	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )
21	6 5 7 8 20	lspsnel5a	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ⊆ { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )
22	7	adantr	⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑊 ∈ LMod )
23	3 6 5	lspsncl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) )
24	23	adantr	⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) )
25		simpr	⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑘 ∈ 𝐾 )
26	3 5	lspsnid	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) )
27	26	adantr	⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) )
28	1 4 2 6	lssvscl	⊢ ( ( ( 𝑊 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) ∧ ( 𝑘 ∈ 𝐾 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) → ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) )
29	22 24 25 27 28	syl22anc	⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) )
30		eleq1a	⊢ ( ( 𝑘 · 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) → ( 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) )
31	29 30	syl	⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑘 ∈ 𝐾 ) → ( 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) )
32	31	rexlimdva	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) → 𝑣 ∈ ( 𝑁 ‘ { 𝑋 } ) ) )
33	32	abssdv	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } ⊆ ( 𝑁 ‘ { 𝑋 } ) )
34	21 33	eqssd	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) = { 𝑣 ∣ ∃ 𝑘 ∈ 𝐾 𝑣 = ( 𝑘 · 𝑋 ) } )