Metamath Proof Explorer

Theorem dihvalcqpre

Description: Value of isomorphism H for a lattice K when -. X .<_ W , given auxiliary atom Q . (Contributed by NM, 6-Mar-2014)

Ref Expression
Hypotheses dihval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihval.l ⊢ ≤ = ( le ‘ 𝐾 )
dihval.j ⊢ ∨ = ( join ‘ 𝐾 )
dihval.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihval.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihval.d ⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
dihval.c ⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
dihval.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihval.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
dihval.p ⊢ ⊕ = ( LSSum ‘ 𝑈 )
Assertion dihvalcqpre ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihval.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihval.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihval.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 dihval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 dihval.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dihval.d ⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
9 dihval.c ⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
10 dihval.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
11 dihval.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
12 dihval.p ⊢ ⊕ = ( LSSum ‘ 𝑈 )
13 11 fvexi ⊢ 𝑆 ∈ V
14 nfv ⊢ Ⅎ 𝑞 ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
15 nfvd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → Ⅎ 𝑞 ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
16 1 2 3 4 5 6 7 8 9 10 11 12 dihvalc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
17 16 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
18 eqeq1 ⊢ ( ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( 𝐼 ‘ 𝑋 ) → ( ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
19 18 adantl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( 𝐼 ‘ 𝑋 ) ) → ( ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ↔ ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
20 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑞 ∈ 𝐴 )
22 simprrl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ¬ 𝑞 ≤ 𝑊 )
23 21 22 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) )
24 simpl3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
25 simpl2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → 𝑋 ∈ 𝐵 )
26 simprrr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
27 simpl3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
28 26 27 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) )
29 1 2 3 4 5 6 8 9 10 12 dihjust ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
30 20 23 24 25 28 29 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
31 30 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝑞 ∈ 𝐴 ∧ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
32 1 2 3 4 5 6 7 8 9 10 11 12 dihlsscpre ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )
33 32 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝑆 )
34 1 2 3 4 5 6 lhpmcvr2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
35 34 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ∃ 𝑞 ∈ 𝐴 ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
36 14 15 17 19 31 33 35 riotasv3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ∧ 𝑆 ∈ V ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
37 13 36 mpan2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )