Theorem h1deot 9467

Description: Membership in orthocomplement of 1-dimensional subspace.

Hypothesis

Ref	Expression
h1deot.1	|- B e. H~

Assertion

Ref	Expression
h1deot	|- (A e. (_|_` {B}) <-> (A e. H~ /\ (A .ih B) = 0))

Proof of Theorem h1deot

Step	Hyp	Ref	Expression
1		h1deot.1	. . . 4 |- B e. H~
2		snssi 2470	. . . 4 |- (B e. H~ -> {B} (_ H~)
3	1, 2	ax-mp 7	. . 3 |- {B} (_ H~
4		ocelt 9149	. . 3 |- ({B} (_ H~ -> (A e. (_|_` {B}) <-> (A e. H~ /\ A.x e. {B} (A .ih x) = 0)))
5	3, 4	ax-mp 7	. 2 |- (A e. (_|_` {B}) <-> (A e. H~ /\ A.x e. {B} (A .ih x) = 0))
6		df-ral 1652	. . . 4 |- (A.x e. {B} (A .ih x) = 0 <-> A.x(x e. {B} -> (A .ih x) = 0))
7		elsn 2425	. . . . . 6 |- (x e. {B} <-> x = B)
8	7	imbi1i 186	. . . . 5 |- ((x e. {B} -> (A .ih x) = 0) <-> (x = B -> (A .ih x) = 0))
9	8	albii 1001	. . . 4 |- (A.x(x e. {B} -> (A .ih x) = 0) <-> A.x(x = B -> (A .ih x) = 0))
10	1	elisseti 1821	. . . . 5 |- B e. V
11		opreq2 3975	. . . . . 6 |- (x = B -> (A .ih x) = (A .ih B))
12	11	eqeq1d 1486	. . . . 5 |- (x = B -> ((A .ih x) = 0 <-> (A .ih B) = 0))
13	10, 12	ceqsalv 1830	. . . 4 |- (A.x(x = B -> (A .ih x) = 0) <-> (A .ih B) = 0)
14	6, 9, 13	3bitr 177	. . 3 |- (A.x e. {B} (A .ih x) = 0 <-> (A .ih B) = 0)
15	14	anbi2i 482	. 2 |- ((A e. H~ /\ A.x e. {B} (A .ih x) = 0) <-> (A e. H~ /\ (A .ih B) = 0))
16	5, 15	bitr 173	1 |- (A e. (_|_` {B}) <-> (A e. H~ /\ (A .ih B) = 0))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  A.wral 1648   (_ wss 2050  {csn 2413  ` cfv 3188  (class class class)co 3969  0cc0 5246  H~chil 8783   .ih csp 8788  _|_cort 8794

This theorem is referenced by:  h1det 9468

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-oc 9119

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