Description: A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012)
Ref | Expression | ||
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Hypotheses | hlsuprexch.b | |- B = ( Base ` K ) |
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hlsuprexch.l | |- .<_ = ( le ` K ) |
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hlsuprexch.j | |- .\/ = ( join ` K ) |
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hlsuprexch.a | |- A = ( Atoms ` K ) |
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Assertion | hlexchb2 | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( Q .\/ X ) <-> ( P .\/ X ) = ( Q .\/ X ) ) ) |
Step | Hyp | Ref | Expression |
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1 | hlsuprexch.b | |- B = ( Base ` K ) |
|
2 | hlsuprexch.l | |- .<_ = ( le ` K ) |
|
3 | hlsuprexch.j | |- .\/ = ( join ` K ) |
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4 | hlsuprexch.a | |- A = ( Atoms ` K ) |
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5 | hlcvl | |- ( K e. HL -> K e. CvLat ) |
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6 | 1 2 3 4 | cvlexchb2 | |- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( Q .\/ X ) <-> ( P .\/ X ) = ( Q .\/ X ) ) ) |
7 | 5 6 | syl3an1 | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( Q .\/ X ) <-> ( P .\/ X ) = ( Q .\/ X ) ) ) |