Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. X and F represent t_1 and f(t) respectively. In their notation, we prove f(t) \/ t_1 = q \/ t_1. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdleme9t.l | |- .<_ = ( le ` K ) |
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| cdleme9t.j | |- .\/ = ( join ` K ) |
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| cdleme9t.m | |- ./\ = ( meet ` K ) |
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| cdleme9t.a | |- A = ( Atoms ` K ) |
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| cdleme9t.h | |- H = ( LHyp ` K ) |
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| cdleme9t.u | |- U = ( ( P .\/ Q ) ./\ W ) |
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| cdleme9t.g | |- F = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) |
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| cdleme9t.x | |- X = ( ( P .\/ T ) ./\ W ) |
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| Assertion | cdleme9tN | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( T e. A /\ -. T .<_ W ) ) /\ -. T .<_ ( P .\/ Q ) ) -> ( F .\/ X ) = ( Q .\/ X ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdleme9t.l | |- .<_ = ( le ` K ) |
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| 2 | cdleme9t.j | |- .\/ = ( join ` K ) |
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| 3 | cdleme9t.m | |- ./\ = ( meet ` K ) |
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| 4 | cdleme9t.a | |- A = ( Atoms ` K ) |
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| 5 | cdleme9t.h | |- H = ( LHyp ` K ) |
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| 6 | cdleme9t.u | |- U = ( ( P .\/ Q ) ./\ W ) |
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| 7 | cdleme9t.g | |- F = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) |
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| 8 | cdleme9t.x | |- X = ( ( P .\/ T ) ./\ W ) |
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| 9 | 1 2 3 4 5 6 7 8 | cdleme9 | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( T e. A /\ -. T .<_ W ) ) /\ -. T .<_ ( P .\/ Q ) ) -> ( F .\/ X ) = ( Q .\/ X ) ) |