Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012)
Ref | Expression | ||
---|---|---|---|
Hypotheses | atcvrj1x.l | |- .<_ = ( le ` K ) |
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atcvrj1x.j | |- .\/ = ( join ` K ) |
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atcvrj1x.c | |- C = ( |
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atcvrj1x.a | |- A = ( Atoms ` K ) |
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Assertion | atcvrj2 | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P C ( Q .\/ R ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atcvrj1x.l | |- .<_ = ( le ` K ) |
|
2 | atcvrj1x.j | |- .\/ = ( join ` K ) |
|
3 | atcvrj1x.c | |- C = ( |
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4 | atcvrj1x.a | |- A = ( Atoms ` K ) |
|
5 | 1 2 3 4 | atcvrj2b | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( Q =/= R /\ P .<_ ( Q .\/ R ) ) <-> P C ( Q .\/ R ) ) ) |
6 | 5 | biimp3a | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P C ( Q .\/ R ) ) |