Description: Part (6) of Baer p. 47 line 6. Note that we use -. X e. ( N{ Y , Z } ) which is equivalent to Baer's "Fx i^i (Fy + Fz)" by lspdisjb . (Convert mapdh6N to use the function HDMap1 .) (Contributed by NM, 17-May-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | hdmap1-6.h | |- H = ( LHyp ` K ) |
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| hdmap1-6.u | |- U = ( ( DVecH ` K ) ` W ) |
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| hdmap1-6.v | |- V = ( Base ` U ) |
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| hdmap1-6.p | |- .+ = ( +g ` U ) |
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| hdmap1-6.o | |- .0. = ( 0g ` U ) |
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| hdmap1-6.n | |- N = ( LSpan ` U ) |
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| hdmap1-6.c | |- C = ( ( LCDual ` K ) ` W ) |
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| hdmap1-6.d | |- D = ( Base ` C ) |
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| hdmap1-6.a | |- .+b = ( +g ` C ) |
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| hdmap1-6.l | |- L = ( LSpan ` C ) |
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| hdmap1-6.m | |- M = ( ( mapd ` K ) ` W ) |
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| hdmap1-6.i | |- I = ( ( HDMap1 ` K ) ` W ) |
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| hdmap1-6.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| hdmap1-6.f | |- ( ph -> F e. D ) |
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| hdmap1-6.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| hdmap1-6.y | |- ( ph -> Y e. V ) |
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| hdmap1-6.z | |- ( ph -> Z e. V ) |
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| hdmap1-6.xn | |- ( ph -> -. X e. ( N ` { Y , Z } ) ) |
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| hdmap1-6.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) ) |
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| Assertion | hdmap1l6 | |- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hdmap1-6.h | |- H = ( LHyp ` K ) |
|
| 2 | hdmap1-6.u | |- U = ( ( DVecH ` K ) ` W ) |
|
| 3 | hdmap1-6.v | |- V = ( Base ` U ) |
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| 4 | hdmap1-6.p | |- .+ = ( +g ` U ) |
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| 5 | hdmap1-6.o | |- .0. = ( 0g ` U ) |
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| 6 | hdmap1-6.n | |- N = ( LSpan ` U ) |
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| 7 | hdmap1-6.c | |- C = ( ( LCDual ` K ) ` W ) |
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| 8 | hdmap1-6.d | |- D = ( Base ` C ) |
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| 9 | hdmap1-6.a | |- .+b = ( +g ` C ) |
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| 10 | hdmap1-6.l | |- L = ( LSpan ` C ) |
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| 11 | hdmap1-6.m | |- M = ( ( mapd ` K ) ` W ) |
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| 12 | hdmap1-6.i | |- I = ( ( HDMap1 ` K ) ` W ) |
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| 13 | hdmap1-6.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
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| 14 | hdmap1-6.f | |- ( ph -> F e. D ) |
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| 15 | hdmap1-6.x | |- ( ph -> X e. ( V \ { .0. } ) ) |
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| 16 | hdmap1-6.y | |- ( ph -> Y e. V ) |
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| 17 | hdmap1-6.z | |- ( ph -> Z e. V ) |
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| 18 | hdmap1-6.xn | |- ( ph -> -. X e. ( N ` { Y , Z } ) ) |
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| 19 | hdmap1-6.mn | |- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) ) |
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| 20 | eqid | |- ( -g ` U ) = ( -g ` U ) |
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| 21 | eqid | |- ( -g ` C ) = ( -g ` C ) |
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| 22 | eqid | |- ( 0g ` C ) = ( 0g ` C ) |
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| 23 | 1 2 3 4 20 5 6 7 8 9 21 22 10 11 12 13 14 15 19 16 17 18 | hdmap1l6k | |- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) |