Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change .\/ order of ( X ./\ Y ) .\/ p here and down? (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dihmeetlem8.b | |- B = ( Base ` K ) |
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| dihmeetlem8.l | |- .<_ = ( le ` K ) |
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| dihmeetlem8.h | |- H = ( LHyp ` K ) |
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| dihmeetlem8.j | |- .\/ = ( join ` K ) |
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| dihmeetlem8.m | |- ./\ = ( meet ` K ) |
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| dihmeetlem8.a | |- A = ( Atoms ` K ) |
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| dihmeetlem8.u | |- U = ( ( DVecH ` K ) ` W ) |
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| dihmeetlem8.s | |- .(+) = ( LSSum ` U ) |
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| dihmeetlem8.i | |- I = ( ( DIsoH ` K ) ` W ) |
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| Assertion | dihmeetlem8N | |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( p e. A /\ -. p .<_ W ) /\ ( p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( ( X ./\ Y ) .\/ p ) ) = ( ( I ` p ) .(+) ( I ` ( X ./\ Y ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dihmeetlem8.b | |- B = ( Base ` K ) |
|
| 2 | dihmeetlem8.l | |- .<_ = ( le ` K ) |
|
| 3 | dihmeetlem8.h | |- H = ( LHyp ` K ) |
|
| 4 | dihmeetlem8.j | |- .\/ = ( join ` K ) |
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| 5 | dihmeetlem8.m | |- ./\ = ( meet ` K ) |
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| 6 | dihmeetlem8.a | |- A = ( Atoms ` K ) |
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| 7 | dihmeetlem8.u | |- U = ( ( DVecH ` K ) ` W ) |
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| 8 | dihmeetlem8.s | |- .(+) = ( LSSum ` U ) |
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| 9 | dihmeetlem8.i | |- I = ( ( DIsoH ` K ) ` W ) |
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| 10 | 1 2 3 4 5 6 7 8 9 | dihjatc1 | |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( p e. A /\ -. p .<_ W ) /\ ( p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( ( X ./\ Y ) .\/ p ) ) = ( ( I ` p ) .(+) ( I ` ( X ./\ Y ) ) ) ) |