Description: Lemma for dath . Planes Y and Z form a 3-dimensional space (when they are different). (Contributed by NM, 22-Jul-2012)
| Ref | Expression | ||
|---|---|---|---|
| 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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| dalemc.l | |- .<_ = ( le ` K ) |
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| dalemc.j | |- .\/ = ( join ` K ) |
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| dalemc.a | |- A = ( Atoms ` K ) |
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| dalem14.o | |- O = ( LPlanes ` K ) |
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| dalem14.v | |- V = ( LVols ` K ) |
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| dalem14.y | |- Y = ( ( P .\/ Q ) .\/ R ) |
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| dalem14.z | |- Z = ( ( S .\/ T ) .\/ U ) |
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| dalem14.w | |- W = ( Y .\/ C ) |
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| Assertion | dalem14 | |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. V ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 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 | dalemc.l | |- .<_ = ( le ` K ) |
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| 3 | dalemc.j | |- .\/ = ( join ` K ) |
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| 4 | dalemc.a | |- A = ( Atoms ` K ) |
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| 5 | dalem14.o | |- O = ( LPlanes ` K ) |
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| 6 | dalem14.v | |- V = ( LVols ` K ) |
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| 7 | dalem14.y | |- Y = ( ( P .\/ Q ) .\/ R ) |
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| 8 | dalem14.z | |- Z = ( ( S .\/ T ) .\/ U ) |
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| 9 | dalem14.w | |- W = ( Y .\/ C ) |
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| 10 | 1 2 3 4 5 7 8 9 | dalem13 | |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) = W ) |
| 11 | 1 2 3 4 5 6 7 8 9 | dalem9 | |- ( ( ph /\ Y =/= Z ) -> W e. V ) |
| 12 | 10 11 | eqeltrd | |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. V ) |