Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)
Ref | Expression | ||
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Hypotheses | dihmeetlem14.b | |- B = ( Base ` K ) |
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dihmeetlem14.l | |- .<_ = ( le ` K ) |
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dihmeetlem14.h | |- H = ( LHyp ` K ) |
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dihmeetlem14.j | |- .\/ = ( join ` K ) |
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dihmeetlem14.m | |- ./\ = ( meet ` K ) |
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dihmeetlem14.a | |- A = ( Atoms ` K ) |
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dihmeetlem14.u | |- U = ( ( DVecH ` K ) ` W ) |
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dihmeetlem14.s | |- .(+) = ( LSSum ` U ) |
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dihmeetlem14.i | |- I = ( ( DIsoH ` K ) ` W ) |
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Assertion | dihmeetlem14N | |- ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ p e. B ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( ( I ` ( Y ./\ p ) ) .(+) ( ( I ` r ) i^i ( I ` p ) ) ) = ( ( I ` Y ) i^i ( I ` p ) ) ) |
Step | Hyp | Ref | Expression |
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1 | dihmeetlem14.b | |- B = ( Base ` K ) |
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2 | dihmeetlem14.l | |- .<_ = ( le ` K ) |
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3 | dihmeetlem14.h | |- H = ( LHyp ` K ) |
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4 | dihmeetlem14.j | |- .\/ = ( join ` K ) |
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5 | dihmeetlem14.m | |- ./\ = ( meet ` K ) |
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6 | dihmeetlem14.a | |- A = ( Atoms ` K ) |
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7 | dihmeetlem14.u | |- U = ( ( DVecH ` K ) ` W ) |
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8 | dihmeetlem14.s | |- .(+) = ( LSSum ` U ) |
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9 | dihmeetlem14.i | |- I = ( ( DIsoH ` K ) ` W ) |
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10 | 1 2 3 4 5 6 7 8 9 | dihmeetlem12N | |- ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ p e. B ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( ( I ` ( Y ./\ p ) ) .(+) ( ( I ` r ) i^i ( I ` p ) ) ) = ( ( I ` Y ) i^i ( I ` p ) ) ) |