Metamath Proof Explorer


Theorem dih2dimb

Description: Extend dib2dim to isomorphism H. (Contributed by NM, 22-Sep-2014)

Ref Expression
Hypotheses dih2dimb.l = ( le ‘ 𝐾 )
dih2dimb.j = ( join ‘ 𝐾 )
dih2dimb.a 𝐴 = ( Atoms ‘ 𝐾 )
dih2dimb.h 𝐻 = ( LHyp ‘ 𝐾 )
dih2dimb.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dih2dimb.s = ( LSSum ‘ 𝑈 )
dih2dimb.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dih2dimb.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
dih2dimb.p ( 𝜑 → ( 𝑃𝐴𝑃 𝑊 ) )
dih2dimb.q ( 𝜑 → ( 𝑄𝐴𝑄 𝑊 ) )
Assertion dih2dimb ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) ⊆ ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 dih2dimb.l = ( le ‘ 𝐾 )
2 dih2dimb.j = ( join ‘ 𝐾 )
3 dih2dimb.a 𝐴 = ( Atoms ‘ 𝐾 )
4 dih2dimb.h 𝐻 = ( LHyp ‘ 𝐾 )
5 dih2dimb.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6 dih2dimb.s = ( LSSum ‘ 𝑈 )
7 dih2dimb.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dih2dimb.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 dih2dimb.p ( 𝜑 → ( 𝑃𝐴𝑃 𝑊 ) )
10 dih2dimb.q ( 𝜑 → ( 𝑄𝐴𝑄 𝑊 ) )
11 eqid ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
12 1 2 3 4 5 6 11 8 9 10 dib2dim ( 𝜑 → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑃 𝑄 ) ) ⊆ ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) )
13 8 simpld ( 𝜑𝐾 ∈ HL )
14 9 simpld ( 𝜑𝑃𝐴 )
15 10 simpld ( 𝜑𝑄𝐴 )
16 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
17 16 2 3 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
18 13 14 15 17 syl3anc ( 𝜑 → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
19 9 simprd ( 𝜑𝑃 𝑊 )
20 10 simprd ( 𝜑𝑄 𝑊 )
21 13 hllatd ( 𝜑𝐾 ∈ Lat )
22 16 3 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
23 14 22 syl ( 𝜑𝑃 ∈ ( Base ‘ 𝐾 ) )
24 16 3 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
25 15 24 syl ( 𝜑𝑄 ∈ ( Base ‘ 𝐾 ) )
26 8 simprd ( 𝜑𝑊𝐻 )
27 16 4 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
28 26 27 syl ( 𝜑𝑊 ∈ ( Base ‘ 𝐾 ) )
29 16 1 2 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 𝑊𝑄 𝑊 ) ↔ ( 𝑃 𝑄 ) 𝑊 ) )
30 21 23 25 28 29 syl13anc ( 𝜑 → ( ( 𝑃 𝑊𝑄 𝑊 ) ↔ ( 𝑃 𝑄 ) 𝑊 ) )
31 19 20 30 mpbi2and ( 𝜑 → ( 𝑃 𝑄 ) 𝑊 )
32 16 1 4 7 11 dihvalb ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 𝑄 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑃 𝑄 ) ) )
33 8 18 31 32 syl12anc ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑃 𝑄 ) ) )
34 16 1 4 7 11 dihvalb ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 𝑊 ) ) → ( 𝐼𝑃 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) )
35 8 23 19 34 syl12anc ( 𝜑 → ( 𝐼𝑃 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) )
36 16 1 4 7 11 dihvalb ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 𝑊 ) ) → ( 𝐼𝑄 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) )
37 8 25 20 36 syl12anc ( 𝜑 → ( 𝐼𝑄 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) )
38 35 37 oveq12d ( 𝜑 → ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) = ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) )
39 12 33 38 3sstr4d ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) ⊆ ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) )