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Metamath Proof Explorer

Theorem dihffval

Description: The 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 )
Assertion 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 ) ) ) ) ) ) ) ) )

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 elex ⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V )
8 fveq2 ⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) )
9 8 6 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 )
10 fveq2 ⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = ( Base ‘ 𝐾 ) )
11 10 1 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = 𝐵 )
12 fveq2 ⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ( le ‘ 𝐾 ) )
13 12 2 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ≤ )
14 13 breqd ⊢ ( 𝑘 = 𝐾 → ( 𝑥 ( le ‘ 𝑘 ) 𝑤 ↔ 𝑥 ≤ 𝑤 ) )
15 fveq2 ⊢ ( 𝑘 = 𝐾 → ( DIsoB ‘ 𝑘 ) = ( DIsoB ‘ 𝐾 ) )
16 15 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) )
17 16 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) )
18 fveq2 ⊢ ( 𝑘 = 𝐾 → ( DVecH ‘ 𝑘 ) = ( DVecH ‘ 𝐾 ) )
19 18 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) )
20 19 fveq2d ⊢ ( 𝑘 = 𝐾 → ( LSubSp ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) )
21 fveq2 ⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = ( Atoms ‘ 𝐾 ) )
22 21 5 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = 𝐴 )
23 13 breqd ⊢ ( 𝑘 = 𝐾 → ( 𝑞 ( le ‘ 𝑘 ) 𝑤 ↔ 𝑞 ≤ 𝑤 ) )
24 23 notbid ⊢ ( 𝑘 = 𝐾 → ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ↔ ¬ 𝑞 ≤ 𝑤 ) )
25 fveq2 ⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ( join ‘ 𝐾 ) )
26 25 3 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ∨ )
27 eqidd ⊢ ( 𝑘 = 𝐾 → 𝑞 = 𝑞 )
28 fveq2 ⊢ ( 𝑘 = 𝐾 → ( meet ‘ 𝑘 ) = ( meet ‘ 𝐾 ) )
29 28 4 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( meet ‘ 𝑘 ) = ∧ )
30 29 oveqd ⊢ ( 𝑘 = 𝐾 → ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) = ( 𝑥 ∧ 𝑤 ) )
31 26 27 30 oveq123d ⊢ ( 𝑘 = 𝐾 → ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) )
32 31 eqeq1d ⊢ ( 𝑘 = 𝐾 → ( ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ↔ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) )
33 24 32 anbi12d ⊢ ( 𝑘 = 𝐾 → ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) ↔ ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) ) )
34 19 fveq2d ⊢ ( 𝑘 = 𝐾 → ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) = ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) )
35 fveq2 ⊢ ( 𝑘 = 𝐾 → ( DIsoC ‘ 𝑘 ) = ( DIsoC ‘ 𝐾 ) )
36 35 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) )
37 36 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) = ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) )
38 16 30 fveq12d ⊢ ( 𝑘 = 𝐾 → ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) )
39 34 37 38 oveq123d ⊢ ( 𝑘 = 𝐾 → ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) )
40 39 eqeq2d ⊢ ( 𝑘 = 𝐾 → ( 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) ) ↔ 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) )
41 33 40 imbi12d ⊢ ( 𝑘 = 𝐾 → ( ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) ) ) ↔ ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) )
42 22 41 raleqbidv ⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) ) ) ↔ ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑤 ∧ ( 𝑞 ∨ ( 𝑥 ∧ 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑥 ∧ 𝑤 ) ) ) ) ) )
43 20 42 riotaeqbidv ⊢ ( 𝑘 = 𝐾 → ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 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 ) ) ) ) ) )
44 14 17 43 ifbieq12d ⊢ ( 𝑘 = 𝐾 → if ( 𝑥 ( le ‘ 𝑘 ) 𝑤 , ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 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 ) ) ) ) ) ) )
45 11 44 mpteq12dv ⊢ ( 𝑘 = 𝐾 → ( 𝑥 ∈ ( Base ‘ 𝑘 ) ↦ if ( 𝑥 ( le ‘ 𝑘 ) 𝑤 , ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 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 ) ) ) ) ) ) ) )
46 9 45 mpteq12dv ⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( 𝑥 ∈ ( Base ‘ 𝑘 ) ↦ if ( 𝑥 ( le ‘ 𝑘 ) 𝑤 , ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 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 ) ) ) ) ) ) ) ) )
47 df-dih ⊢ DIsoH = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( 𝑥 ∈ ( Base ‘ 𝑘 ) ↦ if ( 𝑥 ( le ‘ 𝑘 ) 𝑤 , ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) , ( ℩ 𝑢 ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ( ( ¬ 𝑞 ( le ‘ 𝑘 ) 𝑤 ∧ ( 𝑞 ( join ‘ 𝑘 ) ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑞 ) ( LSSum ‘ ( ( DVecH ‘ 𝑘 ) ‘ 𝑤 ) ) ( ( ( DIsoB ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑥 ( meet ‘ 𝑘 ) 𝑤 ) ) ) ) ) ) ) ) )
48 46 47 6 mptfvmpt ⊢ ( 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 ) ) ) ) ) ) ) ) )
49 7 48 syl ⊢ ( 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 ) ) ) ) ) ) ) ) )