Theorem h1deot 9472

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 2466	. . . 4 |- (B e. H~ -> {B} (_ H~)
3	1, 2	ax-mp 7	. . 3 |- {B} (_ H~
4		ocelt 9154	. . 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 1649	. . . 4 |- (A.x e. {B} (A .ih x) = 0 <-> A.x(x e. {B} -> (A .ih x) = 0))
7		elsn 2421	. . . . . 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 999	. . . 4 |- (A.x(x e. {B} -> (A .ih x) = 0) <-> A.x(x = B -> (A .ih x) = 0))
10	1	elisseti 1818	. . . . 5 |- B e. V
11		opreq2 3969	. . . . . 6 |- (x = B -> (A .ih x) = (A .ih B))
12	11	eqeq1d 1483	. . . . 5 |- (x = B -> ((A .ih x) = 0 <-> (A .ih B) = 0))
13	10, 12	ceqsalv 1827	. . . 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 480	. 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 954   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  {csn 2409  ` cfv 3182  (class class class)co 3963  0cc0 5234  H~chil 8788   .ih csp 8793  _|_cort 8799

This theorem is referenced by:  h1det 9473

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oc 9124

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