Description: Lemmma for ~? mapdh . Part (2) in Baer p. 45. The bidirectional version of mapdheq2 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | mapdh.q | |- Q = ( 0g ` C ) |
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| mapdh.i | |- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) |
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| mapdh.h | |- H = ( LHyp ` K ) |
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| mapdh.m | |- M = ( ( mapd ` K ) ` W ) |
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| mapdh.u | |- U = ( ( DVecH ` K ) ` W ) |
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| mapdh.v | |- V = ( Base ` U ) |
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| mapdh.s | |- .- = ( -g ` U ) |
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| mapdhc.o | |- .0. = ( 0g ` U ) |
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| mapdh.n | |- N = ( LSpan ` U ) |
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| mapdh.c | |- C = ( ( LCDual ` K ) ` W ) |
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| mapdh.d | |- D = ( Base ` C ) |
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| mapdh.r | |- R = ( -g ` C ) |
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| mapdh.j | |- J = ( LSpan ` C ) |
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| mapdh.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| mapdhc.f | |- ( ph -> F e. D ) |
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| mapdh.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
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| mapdhcl.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| mapdhe2.y | |- ( ph -> Y e. ( V \ { .0. } ) ) |
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| mapdhe2.g | |- ( ph -> G e. D ) |
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| mapdh.ne3 | |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
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| mapdh.my | |- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) ) |
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| Assertion | mapdheq2biN | |- ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( I ` <. Y , G , X >. ) = F ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdh.q | |- Q = ( 0g ` C ) |
|
| 2 | mapdh.i | |- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) |
|
| 3 | mapdh.h | |- H = ( LHyp ` K ) |
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| 4 | mapdh.m | |- M = ( ( mapd ` K ) ` W ) |
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| 5 | mapdh.u | |- U = ( ( DVecH ` K ) ` W ) |
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| 6 | mapdh.v | |- V = ( Base ` U ) |
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| 7 | mapdh.s | |- .- = ( -g ` U ) |
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| 8 | mapdhc.o | |- .0. = ( 0g ` U ) |
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| 9 | mapdh.n | |- N = ( LSpan ` U ) |
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| 10 | mapdh.c | |- C = ( ( LCDual ` K ) ` W ) |
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| 11 | mapdh.d | |- D = ( Base ` C ) |
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| 12 | mapdh.r | |- R = ( -g ` C ) |
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| 13 | mapdh.j | |- J = ( LSpan ` C ) |
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| 14 | mapdh.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| 15 | mapdhc.f | |- ( ph -> F e. D ) |
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| 16 | mapdh.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
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| 17 | mapdhcl.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| 18 | mapdhe2.y | |- ( ph -> Y e. ( V \ { .0. } ) ) |
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| 19 | mapdhe2.g | |- ( ph -> G e. D ) |
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| 20 | mapdh.ne3 | |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
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| 21 | mapdh.my | |- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) ) |
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| 22 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | mapdheq2 | |- ( ph -> ( ( I ` <. X , F , Y >. ) = G -> ( I ` <. Y , G , X >. ) = F ) ) |
| 23 | 20 | necomd | |- ( ph -> ( N ` { Y } ) =/= ( N ` { X } ) ) |
| 24 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 19 21 18 17 15 23 | mapdheq2 | |- ( ph -> ( ( I ` <. Y , G , X >. ) = F -> ( I ` <. X , F , Y >. ) = G ) ) |
| 25 | 22 24 | impbid | |- ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( I ` <. Y , G , X >. ) = F ) ) |