Description: TODO FIX COMMENT. (Contributed by NM, 1-Apr-2013) (New usage is discouraged.)
Ref | Expression | ||
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Hypotheses | cdlemef46g.b | |- B = ( Base ` K ) |
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cdlemef46g.l | |- .<_ = ( le ` K ) |
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cdlemef46g.j | |- .\/ = ( join ` K ) |
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cdlemef46g.m | |- ./\ = ( meet ` K ) |
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cdlemef46g.a | |- A = ( Atoms ` K ) |
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cdlemef46g.h | |- H = ( LHyp ` K ) |
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cdlemef46g.u | |- U = ( ( P .\/ Q ) ./\ W ) |
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cdlemef46g.d | |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
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cdlemefs46g.e | |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) |
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cdlemef46g.f | |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) ) |
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cdlemef46.v | |- V = ( ( Q .\/ P ) ./\ W ) |
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cdlemef46.n | |- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) |
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cdlemefs46.o | |- O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) ) |
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cdlemef46.g | |- G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) ) |
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Assertion | cdlemeg46bOLDN | |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( G ` S ) = [_ S / v ]_ N ) |
Step | Hyp | Ref | Expression |
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1 | cdlemef46g.b | |- B = ( Base ` K ) |
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2 | cdlemef46g.l | |- .<_ = ( le ` K ) |
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3 | cdlemef46g.j | |- .\/ = ( join ` K ) |
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4 | cdlemef46g.m | |- ./\ = ( meet ` K ) |
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5 | cdlemef46g.a | |- A = ( Atoms ` K ) |
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6 | cdlemef46g.h | |- H = ( LHyp ` K ) |
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7 | cdlemef46g.u | |- U = ( ( P .\/ Q ) ./\ W ) |
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8 | cdlemef46g.d | |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
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9 | cdlemefs46g.e | |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) |
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10 | cdlemef46g.f | |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) ) |
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11 | cdlemef46.v | |- V = ( ( Q .\/ P ) ./\ W ) |
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12 | cdlemef46.n | |- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) |
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13 | cdlemefs46.o | |- O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) ) |
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14 | cdlemef46.g | |- G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) ) |
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15 | 1 2 3 4 5 6 11 12 13 14 | cdlemeg47b | |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( G ` S ) = [_ S / v ]_ N ) |