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Metamath Proof Explorer

Theorem dihval

Description: Value of isomorphism H for a lattice K . Definition of isomorphism map in Crawley p. 122 line 3. (Contributed by NM, 3-Feb-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 dihval ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) = if ( 𝑋 ≤ 𝑊 , ( 𝐷 ‘ 𝑋 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 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 1 2 3 4 5 6 7 8 9 10 11 12 dihfval ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) ) )
14 13 fveq1d ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) ) ‘ 𝑋 ) )
15 breq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊 ) )
16 fveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝐷 ‘ 𝑥 ) = ( 𝐷 ‘ 𝑋 ) )
17 oveq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∧ 𝑊 ) = ( 𝑋 ∧ 𝑊 ) )
18 17 oveq2d ⊢ ( 𝑥 = 𝑋 → ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) )
19 id ⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 )
20 18 19 eqeq12d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ↔ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
21 20 anbi2d ⊢ ( 𝑥 = 𝑋 → ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) ↔ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) )
22 fvoveq1 ⊢ ( 𝑥 = 𝑋 → ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) = ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) )
23 22 oveq2d ⊢ ( 𝑥 = 𝑋 → ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
24 23 eqeq2d ⊢ ( 𝑥 = 𝑋 → ( 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ↔ 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
25 21 24 imbi12d ⊢ ( 𝑥 = 𝑋 → ( ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ↔ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
26 25 ralbidv ⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ↔ ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
27 26 riotabidv ⊢ ( 𝑥 = 𝑋 → ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
28 15 16 27 ifbieq12d ⊢ ( 𝑥 = 𝑋 → if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) = if ( 𝑋 ≤ 𝑊 , ( 𝐷 ‘ 𝑋 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) )
29 eqid ⊢ ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) ) = ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) )
30 fvex ⊢ ( 𝐷 ‘ 𝑋 ) ∈ V
31 riotaex ⊢ ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ∈ V
32 30 31 ifex ⊢ if ( 𝑋 ≤ 𝑊 , ( 𝐷 ‘ 𝑋 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) ∈ V
33 28 29 32 fvmpt ⊢ ( 𝑋 ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) ) ‘ 𝑋 ) = if ( 𝑋 ≤ 𝑊 , ( 𝐷 ‘ 𝑋 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) )
34 14 33 sylan9eq ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) = if ( 𝑋 ≤ 𝑊 , ( 𝐷 ‘ 𝑋 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) )