Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme3fa and cdleme3 . TODO: Delete - duplicates cdleme0e . (Contributed by NM, 6-Jun-2012) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdleme1.l | |- .<_ = ( le ` K ) |
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| cdleme1.j | |- .\/ = ( join ` K ) |
||
| cdleme1.m | |- ./\ = ( meet ` K ) |
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| cdleme1.a | |- A = ( Atoms ` K ) |
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| cdleme1.h | |- H = ( LHyp ` K ) |
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| cdleme1.u | |- U = ( ( P .\/ Q ) ./\ W ) |
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| cdleme1.f | |- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
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| cdleme3.3 | |- V = ( ( P .\/ R ) ./\ W ) |
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| Assertion | cdleme3fN | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> U =/= V ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdleme1.l | |- .<_ = ( le ` K ) |
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| 2 | cdleme1.j | |- .\/ = ( join ` K ) |
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| 3 | cdleme1.m | |- ./\ = ( meet ` K ) |
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| 4 | cdleme1.a | |- A = ( Atoms ` K ) |
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| 5 | cdleme1.h | |- H = ( LHyp ` K ) |
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| 6 | cdleme1.u | |- U = ( ( P .\/ Q ) ./\ W ) |
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| 7 | cdleme1.f | |- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
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| 8 | cdleme3.3 | |- V = ( ( P .\/ R ) ./\ W ) |
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| 9 | 1 2 3 4 5 6 8 | cdleme0e | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> U =/= V ) |