Description: Part (7) of Baer p. 48 line 10 (5 of 6 cases). X , Y , Z are Baer's u, v, w. (Note: Cases 1 of 6 and 2 of 6 are hypotheses mapdh75b here and mapdh75a in mapdh75cN .) (Contributed by NM, 2-May-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | mapdh75.h | |- H = ( LHyp ` K ) |
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| mapdh75.u | |- U = ( ( DVecH ` K ) ` W ) |
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| mapdh75.v | |- V = ( Base ` U ) |
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| mapdh75.s | |- .- = ( -g ` U ) |
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| mapdh75.o | |- .0. = ( 0g ` U ) |
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| mapdh75.n | |- N = ( LSpan ` U ) |
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| mapdh75.c | |- C = ( ( LCDual ` K ) ` W ) |
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| mapdh75.d | |- D = ( Base ` C ) |
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| mapdh75.r | |- R = ( -g ` C ) |
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| mapdh75.q | |- Q = ( 0g ` C ) |
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| mapdh75.j | |- J = ( LSpan ` C ) |
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| mapdh75.m | |- M = ( ( mapd ` K ) ` W ) |
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| mapdh75.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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| mapdh75.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| mapdh75.f | |- ( ph -> F e. D ) |
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| mapdh75.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
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| mapdh75b | |- ( ph -> ( I ` <. X , F , Z >. ) = E ) |
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| mapdh75e.ne | |- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) ) |
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| mapdh75e.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| mapdh75e.z | |- ( ph -> Z e. ( V \ { .0. } ) ) |
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| Assertion | mapdh75e | |- ( ph -> ( I ` <. Z , E , X >. ) = F ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdh75.h | |- H = ( LHyp ` K ) |
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| 2 | mapdh75.u | |- U = ( ( DVecH ` K ) ` W ) |
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| 3 | mapdh75.v | |- V = ( Base ` U ) |
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| 4 | mapdh75.s | |- .- = ( -g ` U ) |
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| 5 | mapdh75.o | |- .0. = ( 0g ` U ) |
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| 6 | mapdh75.n | |- N = ( LSpan ` U ) |
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| 7 | mapdh75.c | |- C = ( ( LCDual ` K ) ` W ) |
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| 8 | mapdh75.d | |- D = ( Base ` C ) |
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| 9 | mapdh75.r | |- R = ( -g ` C ) |
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| 10 | mapdh75.q | |- Q = ( 0g ` C ) |
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| 11 | mapdh75.j | |- J = ( LSpan ` C ) |
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| 12 | mapdh75.m | |- M = ( ( mapd ` K ) ` W ) |
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| 13 | mapdh75.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 | mapdh75.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| 15 | mapdh75.f | |- ( ph -> F e. D ) |
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| 16 | mapdh75.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
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| 17 | mapdh75b | |- ( ph -> ( I ` <. X , F , Z >. ) = E ) |
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| 18 | mapdh75e.ne | |- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) ) |
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| 19 | mapdh75e.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| 20 | mapdh75e.z | |- ( ph -> Z e. ( V \ { .0. } ) ) |
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| 21 | 20 | eldifad | |- ( ph -> Z e. V ) |
| 22 | 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 21 18 | mapdhcl | |- ( ph -> ( I ` <. X , F , Z >. ) e. D ) |
| 23 | 17 22 | eqeltrrd | |- ( ph -> E e. D ) |
| 24 | 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 20 23 18 | mapdheq2 | |- ( ph -> ( ( I ` <. X , F , Z >. ) = E -> ( I ` <. Z , E , X >. ) = F ) ) |
| 25 | 17 24 | mpd | |- ( ph -> ( I ` <. Z , E , X >. ) = F ) |