Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 13-May-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | mapdh8a.h | |- H = ( LHyp ` K ) |
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| mapdh8a.u | |- U = ( ( DVecH ` K ) ` W ) |
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| mapdh8a.v | |- V = ( Base ` U ) |
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| mapdh8a.s | |- .- = ( -g ` U ) |
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| mapdh8a.o | |- .0. = ( 0g ` U ) |
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| mapdh8a.n | |- N = ( LSpan ` U ) |
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| mapdh8a.c | |- C = ( ( LCDual ` K ) ` W ) |
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| mapdh8a.d | |- D = ( Base ` C ) |
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| mapdh8a.r | |- R = ( -g ` C ) |
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| mapdh8a.q | |- Q = ( 0g ` C ) |
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| mapdh8a.j | |- J = ( LSpan ` C ) |
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| mapdh8a.m | |- M = ( ( mapd ` K ) ` W ) |
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| mapdh8a.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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| mapdh8a.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| mapdh8ac.f | |- ( ph -> F e. D ) |
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| mapdh8ac.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
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| mapdh8ac.eg | |- ( ph -> ( I ` <. X , F , Y >. ) = G ) |
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| mapdh8ac.ee | |- ( ph -> ( I ` <. X , F , Z >. ) = E ) |
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| mapdh8ac.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| mapdh8ac.y | |- ( ph -> Y e. ( V \ { .0. } ) ) |
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| mapdh8ac.z | |- ( ph -> Z e. ( V \ { .0. } ) ) |
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| mapdh8ac.t | |- ( ph -> T e. ( V \ { .0. } ) ) |
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| mapdh8ac.yn | |- ( ph -> ( N ` { X } ) = ( N ` { T } ) ) |
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| mapdh8ac.ew | |- ( ph -> ( I ` <. X , F , w >. ) = B ) |
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| mapdh8ac.w | |- ( ph -> w e. ( V \ { .0. } ) ) |
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| mapdh8ac.yw | |- ( ph -> ( N ` { Y } ) =/= ( N ` { w } ) ) |
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| mapdh8ac.xy | |- ( ph -> -. X e. ( N ` { Y , w } ) ) |
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| mapdh8ac.wz | |- ( ph -> ( N ` { w } ) =/= ( N ` { Z } ) ) |
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| mapdh8ac.xz | |- ( ph -> -. X e. ( N ` { w , Z } ) ) |
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| Assertion | mapdh8ac | |- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. Z , E , T >. ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdh8a.h | |- H = ( LHyp ` K ) |
|
| 2 | mapdh8a.u | |- U = ( ( DVecH ` K ) ` W ) |
|
| 3 | mapdh8a.v | |- V = ( Base ` U ) |
|
| 4 | mapdh8a.s | |- .- = ( -g ` U ) |
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| 5 | mapdh8a.o | |- .0. = ( 0g ` U ) |
|
| 6 | mapdh8a.n | |- N = ( LSpan ` U ) |
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| 7 | mapdh8a.c | |- C = ( ( LCDual ` K ) ` W ) |
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| 8 | mapdh8a.d | |- D = ( Base ` C ) |
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| 9 | mapdh8a.r | |- R = ( -g ` C ) |
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| 10 | mapdh8a.q | |- Q = ( 0g ` C ) |
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| 11 | mapdh8a.j | |- J = ( LSpan ` C ) |
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| 12 | mapdh8a.m | |- M = ( ( mapd ` K ) ` W ) |
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| 13 | mapdh8a.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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| 14 | mapdh8a.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| 15 | mapdh8ac.f | |- ( ph -> F e. D ) |
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| 16 | mapdh8ac.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
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| 17 | mapdh8ac.eg | |- ( ph -> ( I ` <. X , F , Y >. ) = G ) |
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| 18 | mapdh8ac.ee | |- ( ph -> ( I ` <. X , F , Z >. ) = E ) |
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| 19 | mapdh8ac.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| 20 | mapdh8ac.y | |- ( ph -> Y e. ( V \ { .0. } ) ) |
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| 21 | mapdh8ac.z | |- ( ph -> Z e. ( V \ { .0. } ) ) |
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| 22 | mapdh8ac.t | |- ( ph -> T e. ( V \ { .0. } ) ) |
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| 23 | mapdh8ac.yn | |- ( ph -> ( N ` { X } ) = ( N ` { T } ) ) |
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| 24 | mapdh8ac.ew | |- ( ph -> ( I ` <. X , F , w >. ) = B ) |
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| 25 | mapdh8ac.w | |- ( ph -> w e. ( V \ { .0. } ) ) |
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| 26 | mapdh8ac.yw | |- ( ph -> ( N ` { Y } ) =/= ( N ` { w } ) ) |
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| 27 | mapdh8ac.xy | |- ( ph -> -. X e. ( N ` { Y , w } ) ) |
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| 28 | mapdh8ac.wz | |- ( ph -> ( N ` { w } ) =/= ( N ` { Z } ) ) |
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| 29 | mapdh8ac.xz | |- ( ph -> -. X e. ( N ` { w , Z } ) ) |
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| 30 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 24 19 20 25 22 26 27 23 | mapdh8ab | |- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. w , B , T >. ) ) |
| 31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 24 18 19 25 21 22 28 29 23 | mapdh8ab | |- ( ph -> ( I ` <. w , B , T >. ) = ( I ` <. Z , E , T >. ) ) |
| 32 | 30 31 | eqtrd | |- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. Z , E , T >. ) ) |