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Metamath Proof Explorer

Theorem dihvalcq2

Description: Value of isomorphism H for a lattice K when -. X .<_ W , given auxiliary atom Q . (Contributed by NM, 26-Sep-2014)

Ref Expression
Hypotheses dihvalcq2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihvalcq2.l ⊢ ≤ = ( le ‘ 𝐾 )
dihvalcq2.j ⊢ ∨ = ( join ‘ 𝐾 )
dihvalcq2.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihvalcq2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihvalcq2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihvalcq2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihvalcq2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihvalcq2.p ⊢ ⊕ = ( LSSum ‘ 𝑈 )
Assertion dihvalcq2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihvalcq2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihvalcq2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihvalcq2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihvalcq2.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihvalcq2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 dihvalcq2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 dihvalcq2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dihvalcq2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
9 dihvalcq2.p ⊢ ⊕ = ( LSSum ‘ 𝑈 )
10 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
12 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
13 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ≤ 𝑋 )
14 1 2 3 4 5 6 lhpmcvr3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑄 ≤ 𝑋 ↔ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
15 10 11 12 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑄 ≤ 𝑋 ↔ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
16 13 15 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
17 eqid ⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
18 eqid ⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
19 1 2 3 4 5 6 7 17 18 8 9 dihvalcq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
20 10 11 12 16 19 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
21 2 5 6 18 7 dihvalcqat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) )
22 10 12 21 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝐼 ‘ 𝑄 ) = ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) )
23 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ HL )
24 23 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ Lat )
25 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 )
26 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑊 ∈ 𝐻 )
27 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
28 26 27 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑊 ∈ 𝐵 )
29 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
30 24 25 28 29 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
31 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
32 24 25 28 31 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
33 1 2 6 7 17 dihvalb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ∧ 𝑊 ) ) )
34 10 30 32 33 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ∧ 𝑊 ) ) )
35 22 34 oveq12d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
36 20 35 eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )