Metamath Proof Explorer

Theorem dihjat5N

Description: Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015) (New usage is discouraged.)

Ref Expression
Hypotheses dihjat5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihjat5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjat5.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjat5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjat5.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjat5.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjat5.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihjat5.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihjat5.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
dihjat5.p ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
Assertion dihjat5N ( 𝜑 → ( 𝑋 ∨ 𝑃 ) = ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjat5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihjat5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dihjat5.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihjat5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 dihjat5.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6 dihjat5.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
7 dihjat5.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dihjat5.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 dihjat5.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
10 dihjat5.p ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
11 1 2 3 4 5 6 7 8 9 10 dihjat3 ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑋 ∨ 𝑃 ) ) = ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) )
12 eqid ⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 )
13 1 2 7 dihcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )
14 8 9 13 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )
15 4 2 5 7 12 dihatlat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑃 ) ∈ ( LSAtoms ‘ 𝑈 ) )
16 8 10 15 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ ( LSAtoms ‘ 𝑈 ) )
17 2 7 5 6 12 8 14 16 dihsmatrn ⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ∈ ran 𝐼 )
18 2 7 dihcnvid2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ) = ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) )
19 8 17 18 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ) = ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) )
20 11 19 eqtr4d ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑋 ∨ 𝑃 ) ) = ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ) )
21 8 simpld ⊢ ( 𝜑 → 𝐾 ∈ HL )
22 21 hllatd ⊢ ( 𝜑 → 𝐾 ∈ Lat )
23 1 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
24 10 23 syl ⊢ ( 𝜑 → 𝑃 ∈ 𝐵 )
25 1 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 )
26 22 9 24 25 syl3anc ⊢ ( 𝜑 → ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 )
27 1 2 7 dihcnvcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ∈ 𝐵 )
28 8 17 27 syl2anc ⊢ ( 𝜑 → ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ∈ 𝐵 )
29 1 2 7 dih11 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 ∧ ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ∈ 𝐵 ) → ( ( 𝐼 ‘ ( 𝑋 ∨ 𝑃 ) ) = ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ) ↔ ( 𝑋 ∨ 𝑃 ) = ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ) )
30 8 26 28 29 syl3anc ⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝑋 ∨ 𝑃 ) ) = ( 𝐼 ‘ ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ) ↔ ( 𝑋 ∨ 𝑃 ) = ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) ) )
31 20 30 mpbid ⊢ ( 𝜑 → ( 𝑋 ∨ 𝑃 ) = ( ◡ 𝐼 ‘ ( ( 𝐼 ‘ 𝑋 ) ⊕ ( 𝐼 ‘ 𝑃 ) ) ) )