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 ) |
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| dihvalcq.l | |- .<_ = ( le ` K ) |
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| dihvalcq.j | |- .\/ = ( join ` K ) |
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| dihvalcq.m | |- ./\ = ( meet ` K ) |
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| dihvalcq.a | |- A = ( Atoms ` K ) |
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| dihvalcq.h | |- H = ( LHyp ` K ) |
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| dihvalcq.i | |- I = ( ( DIsoH ` K ) ` W ) |
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| dihvalcq.d | |- D = ( ( DIsoB ` K ) ` W ) |
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| dihvalcq.c | |- C = ( ( DIsoC ` K ) ` W ) |
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| dihvalcq.u | |- U = ( ( DVecH ` K ) ` W ) |
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| dihvalcq.p | |- .(+) = ( LSSum ` U ) |
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| Assertion | dihvalcq | |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) ) -> ( I ` X ) = ( ( C ` Q ) .(+) ( D ` ( X ./\ W ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dihvalcq.b | |- B = ( Base ` K ) |
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| 2 | dihvalcq.l | |- .<_ = ( le ` K ) |
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| 3 | dihvalcq.j | |- .\/ = ( join ` K ) |
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| 4 | dihvalcq.m | |- ./\ = ( meet ` K ) |
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| 5 | dihvalcq.a | |- A = ( Atoms ` K ) |
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| 6 | dihvalcq.h | |- H = ( LHyp ` K ) |
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| 7 | dihvalcq.i | |- I = ( ( DIsoH ` K ) ` W ) |
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| 8 | dihvalcq.d | |- D = ( ( DIsoB ` K ) ` W ) |
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| 9 | dihvalcq.c | |- C = ( ( DIsoC ` K ) ` W ) |
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| 10 | dihvalcq.u | |- U = ( ( DVecH ` K ) ` W ) |
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| 11 | dihvalcq.p | |- .(+) = ( LSSum ` U ) |
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| 12 | eqid | |- ( LSubSp ` U ) = ( LSubSp ` U ) |
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| 13 | 1 2 3 4 5 6 7 8 9 10 12 11 | dihvalcqpre | |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) ) -> ( I ` X ) = ( ( C ` Q ) .(+) ( D ` ( X ./\ W ) ) ) ) |