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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dalemb.j | |- .\/ = ( join ` K ) |
||
dalemb.a | |- A = ( Atoms ` K ) |
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Assertion | dalemtjueb | |- ( ph -> ( T .\/ U ) 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 | dalemb.j | |- .\/ = ( join ` K ) |
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3 | dalemb.a | |- A = ( Atoms ` K ) |
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4 | 1 | dalemkehl | |- ( ph -> K e. HL ) |
5 | 1 | dalemtea | |- ( ph -> T e. A ) |
6 | 1 | dalemuea | |- ( ph -> U e. A ) |
7 | eqid | |- ( Base ` K ) = ( Base ` K ) |
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8 | 7 2 3 | hlatjcl | |- ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) ) |
9 | 4 5 6 8 | syl3anc | |- ( ph -> ( T .\/ U ) e. ( Base ` K ) ) |