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Metamath Proof Explorer

Theorem dihlss

Description: The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014)

Ref Expression
Hypotheses dihlss.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihlss.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihlss.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihlss.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihlss.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
Assertion dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 dihlss.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihlss.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dihlss.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 dihlss.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dihlss.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
6 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7 eqid ⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8 1 6 2 3 7 dihvalb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
9 1 6 2 4 7 5 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ∈ 𝑆 )
10 8 9 eqeltrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )
11 10 anassrs ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )
12 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
13 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
14 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
15 eqid ⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
16 eqid ⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
17 1 6 12 13 14 2 3 7 15 4 5 16 dihlsscpre ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )
18 17 anassrs ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )
19 11 18 pm2.61dan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )