Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012)
Ref | Expression | ||
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Hypotheses | dalema.ph | |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) ) |
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dalema.a | |- A = ( Atoms ` K ) |
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Assertion | dalemreb | |- ( ph -> R e. ( Base ` K ) ) |
Step | Hyp | Ref | Expression |
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1 | dalema.ph | |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) ) |
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2 | dalema.a | |- A = ( Atoms ` K ) |
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3 | 1 | dalemrea | |- ( ph -> R e. A ) |
4 | eqid | |- ( Base ` K ) = ( Base ` K ) |
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5 | 4 2 | atbase | |- ( R e. A -> R e. ( Base ` K ) ) |
6 | 3 5 | syl | |- ( ph -> R e. ( Base ` K ) ) |