Description: Value of isomorphism H for a lattice K when -. X .<_ W , given auxiliary atom Q . TODO: Use dihvalcq2 (with lhpmcvr3 for ( Q .\/ ( X ./\ W ) ) = X simplification) that changes C and D to I and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dihvalcq.b | ⊢ 𝐵 = ( Base ‘ 𝐾 ) | |
| dihvalcq.l | ⊢ ≤ = ( le ‘ 𝐾 ) | ||
| dihvalcq.j | ⊢ ∨ = ( join ‘ 𝐾 ) | ||
| dihvalcq.m | ⊢ ∧ = ( meet ‘ 𝐾 ) | ||
| dihvalcq.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | ||
| dihvalcq.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | ||
| dihvalcq.i | ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) | ||
| dihvalcq.d | ⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) | ||
| dihvalcq.c | ⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) | ||
| dihvalcq.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | ||
| dihvalcq.p | ⊢ ⊕ = ( LSSum ‘ 𝑈 ) | ||
| Assertion | dihvalcq | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dihvalcq.b | ⊢ 𝐵 = ( Base ‘ 𝐾 ) | |
| 2 | dihvalcq.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
| 3 | dihvalcq.j | ⊢ ∨ = ( join ‘ 𝐾 ) | |
| 4 | dihvalcq.m | ⊢ ∧ = ( meet ‘ 𝐾 ) | |
| 5 | dihvalcq.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | |
| 6 | dihvalcq.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| 7 | dihvalcq.i | ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) | |
| 8 | dihvalcq.d | ⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) | |
| 9 | dihvalcq.c | ⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) | |
| 10 | dihvalcq.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | |
| 11 | dihvalcq.p | ⊢ ⊕ = ( LSSum ‘ 𝑈 ) | |
| 12 | eqid | ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) | |
| 13 | 1 2 3 4 5 6 7 8 9 10 12 11 | dihvalcqpre | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐶 ‘ 𝑄 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |