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Metamath Proof Explorer

Theorem dihfval

Description: Isomorphism H for a lattice K . Definition of isomorphism map in Crawley p. 122 line 3. (Contributed by NM, 28-Jan-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 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 ) ) ) ) ) ) ) )

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 dihffval ⊢ ( 𝐾 ∈ 𝑉 → ( DIsoH ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) ) ) ) )
14 13 fveq1d ⊢ ( 𝐾 ∈ 𝑉 → ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) ) ) ) ‘ 𝑊 ) )
15 7 14 eqtrid ⊢ ( 𝐾 ∈ 𝑉 → 𝐼 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) ) ) ) ‘ 𝑊 ) )
16 breq2 ⊢ ( 𝑤 = 𝑊 → ( 𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊 ) )
17 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) )
18 17 8 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) = 𝐷 )
19 18 fveq1d ⊢ ( 𝑤 = 𝑊 → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) = ( 𝐷 ‘ 𝑥 ) )
20 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
21 20 10 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = 𝑈 )
22 21 fveq2d ⊢ ( 𝑤 = 𝑊 → ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LSubSp ‘ 𝑈 ) )
23 22 11 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑆 )
24 breq2 ⊢ ( 𝑤 = 𝑊 → ( 𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊 ) )
25 24 notbid ⊢ ( 𝑤 = 𝑊 → ( ¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊 ) )
26 oveq2 ⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∧ 𝑤 ) = ( 𝑥 ∧ 𝑊 ) )
27 26 oveq2d ⊢ ( 𝑤 = 𝑊 → ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) )
28 27 eqeq1d ⊢ ( 𝑤 = 𝑊 → ( ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ↔ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) )
29 25 28 anbi12d ⊢ ( 𝑤 = 𝑊 → ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) ↔ ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) ) )
30 21 fveq2d ⊢ ( 𝑤 = 𝑊 → ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LSSum ‘ 𝑈 ) )
31 30 12 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ⊕ )
32 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) )
33 32 9 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) = 𝐶 )
34 33 fveq1d ⊢ ( 𝑤 = 𝑊 → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) = ( 𝐶 ‘ 𝑞 ) )
35 18 26 fveq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) = ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) )
36 31 34 35 oveq123d ⊢ ( 𝑤 = 𝑊 → ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) )
37 36 eqeq2d ⊢ ( 𝑤 = 𝑊 → ( 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ↔ 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) )
38 29 37 imbi12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ↔ ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) )
39 38 ralbidv ⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ↔ ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) )
40 23 39 riotaeqbidv ⊢ ( 𝑤 = 𝑊 → ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) )
41 16 19 40 ifbieq12d ⊢ ( 𝑤 = 𝑊 → if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) ) = if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) )
42 41 mpteq2dv ⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 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 ) ) ) ) ) ) ) )
43 eqid ⊢ ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) ) ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) ) ) )
44 42 43 1 mptfvmpt ⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑤 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) ) ) ) ‘ 𝑊 ) = ( 𝑥 ∈ 𝐵 ↦ if ( 𝑥 ≤ 𝑊 , ( 𝐷 ‘ 𝑥 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑥 ∧ 𝑊 ) ) ) ) ) ) ) )
45 15 44 sylan9eq ⊢ ( ( 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 ) ) ) ) ) ) ) )