Description: Lemma for dath . Combine the cases where Y and Z are different planes with the case where Y and Z are the same plane. (Contributed by NM, 11-Aug-2012)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dalem62.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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| dalem62.l | |- .<_ = ( le ` K ) |
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| dalem62.j | |- .\/ = ( join ` K ) |
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| dalem62.a | |- A = ( Atoms ` K ) |
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| dalem62.m | |- ./\ = ( meet ` K ) |
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| dalem62.o | |- O = ( LPlanes ` K ) |
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| dalem62.y | |- Y = ( ( P .\/ Q ) .\/ R ) |
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| dalem62.z | |- Z = ( ( S .\/ T ) .\/ U ) |
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| dalem62.d | |- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) ) |
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| dalem62.e | |- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) ) |
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| dalem62.f | |- F = ( ( R .\/ P ) ./\ ( U .\/ S ) ) |
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| Assertion | dalem63 | |- ( ph -> F .<_ ( D .\/ E ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dalem62.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 | dalem62.l | |- .<_ = ( le ` K ) |
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| 3 | dalem62.j | |- .\/ = ( join ` K ) |
|
| 4 | dalem62.a | |- A = ( Atoms ` K ) |
|
| 5 | dalem62.m | |- ./\ = ( meet ` K ) |
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| 6 | dalem62.o | |- O = ( LPlanes ` K ) |
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| 7 | dalem62.y | |- Y = ( ( P .\/ Q ) .\/ R ) |
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| 8 | dalem62.z | |- Z = ( ( S .\/ T ) .\/ U ) |
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| 9 | dalem62.d | |- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) ) |
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| 10 | dalem62.e | |- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) ) |
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| 11 | dalem62.f | |- F = ( ( R .\/ P ) ./\ ( U .\/ S ) ) |
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| 12 | 1 2 3 4 5 6 7 8 9 10 11 | dalem62 | |- ( ( ph /\ Y = Z ) -> F .<_ ( D .\/ E ) ) |
| 13 | 1 2 3 4 5 6 7 8 9 10 11 | dalem16 | |- ( ( ph /\ Y =/= Z ) -> F .<_ ( D .\/ E ) ) |
| 14 | 12 13 | pm2.61dane | |- ( ph -> F .<_ ( D .\/ E ) ) |