Metamath Proof Explorer


Theorem lcfrvalsnN

Description: Reconstruction from the dual space span of a singleton. (Contributed by NM, 19-Feb-2015) (New usage is discouraged.)

Ref Expression
Hypotheses lcfrvalsn.h 𝐻 = ( LHyp ‘ 𝐾 )
lcfrvalsn.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lcfrvalsn.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lcfrvalsn.f 𝐹 = ( LFnl ‘ 𝑈 )
lcfrvalsn.l 𝐿 = ( LKer ‘ 𝑈 )
lcfrvalsn.d 𝐷 = ( LDual ‘ 𝑈 )
lcfrvalsn.n 𝑁 = ( LSpan ‘ 𝐷 )
lcfrvalsn.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
lcfrvalsn.g ( 𝜑𝐺𝐹 )
lcfrvalsn.q 𝑄 = 𝑓𝑅 ( ‘ ( 𝐿𝑓 ) )
lcfrvalsn.r 𝑅 = ( 𝑁 ‘ { 𝐺 } )
Assertion lcfrvalsnN ( 𝜑𝑄 = ( ‘ ( 𝐿𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 lcfrvalsn.h 𝐻 = ( LHyp ‘ 𝐾 )
2 lcfrvalsn.o = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 lcfrvalsn.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 lcfrvalsn.f 𝐹 = ( LFnl ‘ 𝑈 )
5 lcfrvalsn.l 𝐿 = ( LKer ‘ 𝑈 )
6 lcfrvalsn.d 𝐷 = ( LDual ‘ 𝑈 )
7 lcfrvalsn.n 𝑁 = ( LSpan ‘ 𝐷 )
8 lcfrvalsn.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 lcfrvalsn.g ( 𝜑𝐺𝐹 )
10 lcfrvalsn.q 𝑄 = 𝑓𝑅 ( ‘ ( 𝐿𝑓 ) )
11 lcfrvalsn.r 𝑅 = ( 𝑁 ‘ { 𝐺 } )
12 eliun ( 𝑥 𝑓𝑅 ( ‘ ( 𝐿𝑓 ) ) ↔ ∃ 𝑓𝑅 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) )
13 11 eleq2i ( 𝑓𝑅𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) )
14 8 adantr ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
15 eqid ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
16 1 3 8 dvhlmod ( 𝜑𝑈 ∈ LMod )
17 16 adantr ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑈 ∈ LMod )
18 6 16 lduallmod ( 𝜑𝐷 ∈ LMod )
19 eqid ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
20 4 6 19 16 9 ldualelvbase ( 𝜑𝐺 ∈ ( Base ‘ 𝐷 ) )
21 eqid ( LSubSp ‘ 𝐷 ) = ( LSubSp ‘ 𝐷 )
22 19 21 7 lspsncl ( ( 𝐷 ∈ LMod ∧ 𝐺 ∈ ( Base ‘ 𝐷 ) ) → ( 𝑁 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐷 ) )
23 18 20 22 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐷 ) )
24 19 21 lssel ( ( ( 𝑁 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐷 ) ∧ 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑓 ∈ ( Base ‘ 𝐷 ) )
25 23 24 sylan ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑓 ∈ ( Base ‘ 𝐷 ) )
26 4 6 19 16 ldualvbase ( 𝜑 → ( Base ‘ 𝐷 ) = 𝐹 )
27 26 adantr ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( Base ‘ 𝐷 ) = 𝐹 )
28 25 27 eleqtrd ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑓𝐹 )
29 15 4 5 17 28 lkrssv ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝐿𝑓 ) ⊆ ( Base ‘ 𝑈 ) )
30 eqid ( Scalar ‘ 𝐷 ) = ( Scalar ‘ 𝐷 )
31 eqid ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝐷 ) )
32 eqid ( ·𝑠𝐷 ) = ( ·𝑠𝐷 )
33 30 31 19 32 7 lspsnel ( ( 𝐷 ∈ LMod ∧ 𝐺 ∈ ( Base ‘ 𝐷 ) ) → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝑓 = ( 𝑘 ( ·𝑠𝐷 ) 𝐺 ) ) )
34 18 20 33 syl2anc ( 𝜑 → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝑓 = ( 𝑘 ( ·𝑠𝐷 ) 𝐺 ) ) )
35 eqid ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
36 eqid ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
37 35 36 6 30 31 16 ldualsbase ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
38 37 rexeqdv ( 𝜑 → ( ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝑓 = ( 𝑘 ( ·𝑠𝐷 ) 𝐺 ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠𝐷 ) 𝐺 ) ) )
39 34 38 bitrd ( 𝜑 → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠𝐷 ) 𝐺 ) ) )
40 39 biimpa ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠𝐷 ) 𝐺 ) )
41 1 3 8 dvhlvec ( 𝜑𝑈 ∈ LVec )
42 41 adantr ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝑈 ∈ LVec )
43 9 adantr ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → 𝐺𝐹 )
44 35 36 4 5 6 32 42 43 28 lkrss2N ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( ( 𝐿𝐺 ) ⊆ ( 𝐿𝑓 ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑓 = ( 𝑘 ( ·𝑠𝐷 ) 𝐺 ) ) )
45 40 44 mpbird ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝐿𝐺 ) ⊆ ( 𝐿𝑓 ) )
46 1 3 15 2 dochss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐿𝑓 ) ⊆ ( Base ‘ 𝑈 ) ∧ ( 𝐿𝐺 ) ⊆ ( 𝐿𝑓 ) ) → ( ‘ ( 𝐿𝑓 ) ) ⊆ ( ‘ ( 𝐿𝐺 ) ) )
47 14 29 45 46 syl3anc ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( ‘ ( 𝐿𝑓 ) ) ⊆ ( ‘ ( 𝐿𝐺 ) ) )
48 47 sseld ( ( 𝜑𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) ) → ( 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) → 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) )
49 48 ex ( 𝜑 → ( 𝑓 ∈ ( 𝑁 ‘ { 𝐺 } ) → ( 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) → 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) ) )
50 13 49 syl5bi ( 𝜑 → ( 𝑓𝑅 → ( 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) → 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) ) )
51 50 rexlimdv ( 𝜑 → ( ∃ 𝑓𝑅 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) → 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) )
52 19 7 lspsnid ( ( 𝐷 ∈ LMod ∧ 𝐺 ∈ ( Base ‘ 𝐷 ) ) → 𝐺 ∈ ( 𝑁 ‘ { 𝐺 } ) )
53 18 20 52 syl2anc ( 𝜑𝐺 ∈ ( 𝑁 ‘ { 𝐺 } ) )
54 53 11 eleqtrrdi ( 𝜑𝐺𝑅 )
55 2fveq3 ( 𝑓 = 𝐺 → ( ‘ ( 𝐿𝑓 ) ) = ( ‘ ( 𝐿𝐺 ) ) )
56 55 eleq2d ( 𝑓 = 𝐺 → ( 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) ↔ 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) )
57 56 rspcev ( ( 𝐺𝑅𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) → ∃ 𝑓𝑅 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) )
58 54 57 sylan ( ( 𝜑𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) → ∃ 𝑓𝑅 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) )
59 58 ex ( 𝜑 → ( 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) → ∃ 𝑓𝑅 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) ) )
60 51 59 impbid ( 𝜑 → ( ∃ 𝑓𝑅 𝑥 ∈ ( ‘ ( 𝐿𝑓 ) ) ↔ 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) )
61 12 60 syl5bb ( 𝜑 → ( 𝑥 𝑓𝑅 ( ‘ ( 𝐿𝑓 ) ) ↔ 𝑥 ∈ ( ‘ ( 𝐿𝐺 ) ) ) )
62 61 eqrdv ( 𝜑 𝑓𝑅 ( ‘ ( 𝐿𝑓 ) ) = ( ‘ ( 𝐿𝐺 ) ) )
63 10 62 syl5eq ( 𝜑𝑄 = ( ‘ ( 𝐿𝐺 ) ) )