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Metamath Proof Explorer

Theorem dihlspsnssN

Description: A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dih1dor0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dih1dor0.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihldor0.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
dih1dor0.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
dih1dor0.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
dih1dor0.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihlspsnssN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑇 ∈ 𝑆 ↔ 𝑇 ∈ ran 𝐼 ) )

Proof

Step Hyp Ref Expression
1 dih1dor0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dih1dor0.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dihldor0.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 dih1dor0.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
5 dih1dor0.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
6 dih1dor0.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
7 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = ( 𝑁 ‘ { 𝑋 } ) ) → 𝑇 = ( 𝑁 ‘ { 𝑋 } ) )
8 1 2 3 5 6 dihlsprn ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 )
9 8 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 )
10 9 ad2antrr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ran 𝐼 )
11 7 10 eqeltrd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = ( 𝑁 ‘ { 𝑋 } ) ) → 𝑇 ∈ ran 𝐼 )
12 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → 𝑇 = { ( 0g ‘ 𝑈 ) } )
13 simpll1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
15 eqid ⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
16 14 1 6 2 15 dih0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) = { ( 0g ‘ 𝑈 ) } )
17 13 16 syl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) = { ( 0g ‘ 𝑈 ) } )
18 12 17 eqtr4d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → 𝑇 = ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) )
19 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
20 19 1 6 dihfn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn ( Base ‘ 𝐾 ) )
21 13 20 syl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → 𝐼 Fn ( Base ‘ 𝐾 ) )
22 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → 𝐾 ∈ HL )
23 22 ad2antrr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → 𝐾 ∈ HL )
24 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
25 19 14 op0cl ⊢ ( 𝐾 ∈ OP → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
26 23 24 25 3syl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) )
27 fnfvelrn ⊢ ( ( 𝐼 Fn ( Base ‘ 𝐾 ) ∧ ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) ∈ ran 𝐼 )
28 21 26 27 syl2anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) ∈ ran 𝐼 )
29 18 28 eqeltrd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) ∧ 𝑇 = { ( 0g ‘ 𝑈 ) } ) → 𝑇 ∈ ran 𝐼 )
30 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
31 1 2 30 dvhlvec ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) → 𝑈 ∈ LVec )
32 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) → 𝑇 ∈ 𝑆 )
33 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) → 𝑋 ∈ 𝑉 )
34 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) → 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) )
35 3 15 4 5 lspsnat ⊢ ( ( ( 𝑈 ∈ LVec ∧ 𝑇 ∈ 𝑆 ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑇 = ( 𝑁 ‘ { 𝑋 } ) ∨ 𝑇 = { ( 0g ‘ 𝑈 ) } ) )
36 31 32 33 34 35 syl31anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) → ( 𝑇 = ( 𝑁 ‘ { 𝑋 } ) ∨ 𝑇 = { ( 0g ‘ 𝑈 ) } ) )
37 11 29 36 mpjaodan ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑇 ∈ 𝑆 ) → 𝑇 ∈ ran 𝐼 )
38 37 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑇 ∈ 𝑆 → 𝑇 ∈ ran 𝐼 ) )
39 1 2 6 4 dihsslss ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ran 𝐼 ⊆ 𝑆 )
40 39 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → ran 𝐼 ⊆ 𝑆 )
41 40 sseld ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑇 ∈ ran 𝐼 → 𝑇 ∈ 𝑆 ) )
42 38 41 impbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ ( 𝑁 ‘ { 𝑋 } ) ) → ( 𝑇 ∈ 𝑆 ↔ 𝑇 ∈ ran 𝐼 ) )