Metamath Proof Explorer


Theorem mapdh7eN

Description: Part (7) of Baer p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (Contributed by NM, 2-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh7.h H = LHyp K
mapdh7.u U = DVecH K W
mapdh7.v V = Base U
mapdh7.s - ˙ = - U
mapdh7.o 0 ˙ = 0 U
mapdh7.n N = LSpan U
mapdh7.c C = LCDual K W
mapdh7.d D = Base C
mapdh7.r R = - C
mapdh7.q Q = 0 C
mapdh7.j J = LSpan C
mapdh7.m M = mapd K W
mapdh7.i I = x V if 2 nd x = 0 ˙ Q ι h D | M N 2 nd x = J h M N 1 st 1 st x - ˙ 2 nd x = J 2 nd 1 st x R h
mapdh7.k φ K HL W H
mapdh7.f φ F D
mapdh7.mn φ M N u = J F
mapdh7.x φ u V 0 ˙
mapdh7.y φ v V 0 ˙
mapdh7.z φ w V 0 ˙
mapdh7.ne φ N u N v
mapdh7.wn φ ¬ w N u v
mapdh7b φ I u F w = E
Assertion mapdh7eN φ I w E u = F

Proof

Step Hyp Ref Expression
1 mapdh7.h H = LHyp K
2 mapdh7.u U = DVecH K W
3 mapdh7.v V = Base U
4 mapdh7.s - ˙ = - U
5 mapdh7.o 0 ˙ = 0 U
6 mapdh7.n N = LSpan U
7 mapdh7.c C = LCDual K W
8 mapdh7.d D = Base C
9 mapdh7.r R = - C
10 mapdh7.q Q = 0 C
11 mapdh7.j J = LSpan C
12 mapdh7.m M = mapd K W
13 mapdh7.i I = x V if 2 nd x = 0 ˙ Q ι h D | M N 2 nd x = J h M N 1 st 1 st x - ˙ 2 nd x = J 2 nd 1 st x R h
14 mapdh7.k φ K HL W H
15 mapdh7.f φ F D
16 mapdh7.mn φ M N u = J F
17 mapdh7.x φ u V 0 ˙
18 mapdh7.y φ v V 0 ˙
19 mapdh7.z φ w V 0 ˙
20 mapdh7.ne φ N u N v
21 mapdh7.wn φ ¬ w N u v
22 mapdh7b φ I u F w = E
23 19 eldifad φ w V
24 1 2 14 dvhlvec φ U LVec
25 17 eldifad φ u V
26 18 eldifad φ v V
27 3 6 24 23 25 26 21 lspindpi φ N w N u N w N v
28 27 simpld φ N w N u
29 28 necomd φ N u N w
30 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 23 29 mapdhcl φ I u F w D
31 22 30 eqeltrrd φ E D
32 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 19 31 29 mapdheq2 φ I u F w = E I w E u = F
33 22 32 mpd φ I w E u = F