mmtheorems113 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17411

Statement List for Metamath Proof Explorer - 11201-11300 - Page 113 of 175

Type	Label	Description
Statement
Theorem	cm2ji 11201	A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A C_H B   &   |- A C_H C   =>   |- A C_H (B vH C)
Theorem	cm2mi 11202	A lattice element that commutes with two others also commutes with their meet. Theorem 4.2 of [Beran] p. 49.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A C_H B   &   |- A C_H C   =>   |- A C_H (B i^i C)
Quantum Logic Explorer axioms
Theorem	qlax1i 11203	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.)
|- A e. CH   =>   |- A = (_|_` (_|_` A))
Theorem	qlax2i 11204	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   =>   |- (A vH B) = (B vH A)
Theorem	qlax3i 11205	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- ((A vH B) vH C) = (A vH (B vH C))
Theorem	qlax4i 11206	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   =>   |- (A vH (B vH (_|_` B))) = (B vH (_|_` B))
Theorem	qlax5i 11207	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   =>   |- (A vH (_|_` ((_|_` A) vH B))) = A
Theorem	qlaxr1i 11208	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- A = B   =>   |- B = A
Theorem	qlaxr2i 11209	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A = B   &   |- B = C   =>   |- A = C
Theorem	qlaxr4i 11210	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- A = B   =>   |- (_|_` A) = (_|_` B)
Theorem	qlaxr5i 11211	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A = B   =>   |- (A vH C) = (B vH C)
Theorem	qlaxr3i 11212	A variation of the orthomodular law, showing CH is an orthomodular lattice. (This corresponds to axiom "ax-r3" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- (C vH (_|_` C)) = ((_|_` ((_|_` A) vH (_|_` B))) vH (_|_` (A vH B)))   =>   |- A = B
Orthogonal subspaces
Theorem	osumlem1 11213	Lemma for osumi 11221.
Theorem	osumlem2 11214	Lemma for osumi 11221.
Theorem	osumlem3 11215	Lemma for osumi 11221.
Theorem	osumlem4 11216	Lemma for osumi 11221.
Theorem	osumlem5 11217	Lemma for osumi 11221.
Theorem	osumlem6 11218	Lemma for osumi 11221.
Theorem	osumlem7 11219	Lemma for osumi 11221.
Theorem	osumlem8 11220	Lemma for osumi 11221.
Theorem	osumi 11221	If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. Note that the Axiom of Choice is used for this proof (in osumlem5 11217 and via pjpjth 10891 in osumlem7 11219).
|- A e. CH   &   |- B e. CH   =>   |- (A C_ (_|_` B) -> (A +H B) = (A vH B))
Theorem	osumcori 11222	Corollary of osumi 11221.
|- A e. CH   &   |- B e. CH   =>   |- ((A i^i B) +H (A i^i (_|_` B))) = ((A i^i B) vH (A i^i (_|_` B)))
Theorem	osum 11223	If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67.
|- ((A e. CH /\ B e. CH /\ A C_ (_|_` B)) -> (A +H B) = (A vH B))
Theorem	chso 11224	The subspace sum of a closed subspace and its complement is all of Hilbert space.
|- (A e. CH -> (A +H (_|_` A)) = ~H)
Theorem	osumcor2i 11225	Corollary of osumi 11221, showing it holds under the weaker hypothesis that A and B commute.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B -> (A +H B) = (A vH B))
Theorem	spansnji 11226	The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Proof suggested by Eric Schechter 1-Jun-2004.)
|- A e. CH   &   |- B e. ~H   =>   |- (A +H (span` {B})) = (A vH (span` {B}))
Theorem	spansnj 11227	The subspace sum of a closed subspace and a one-dimensional subspace equals their join.
|- ((A e. CH /\ B e. ~H) -> (A +H (span` {B})) = (A vH (span` {B})))
Theorem	spansnscl 11228	The subspace sum of a closed subspace and a one-dimensional subspace is closed.
|- ((A e. CH /\ B e. ~H) -> (A +H (span` {B})) e. CH)
Theorem	sumspansn 11229	The sum of two vectors belong to the span of one of them iff the other vector also belongs.
|- ((A e. ~H /\ B e. ~H) -> ((A +h B) e. (span` {A}) <-> B e. (span` {A})))
Theorem	spansnm0i 11230	The meet of different one-dimensional subspaces is the zero subspace.
|- A e. ~H   &   |- B e. ~H   =>   |- (-. A e. (span` {B}) -> ((span` {A}) i^i (span` {B})) = 0H)
Theorem	nonbooli 11231	A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where ((H i^i F) vH (H i^i G)) = 0H but (H i^i (F vH G)) =/= 0H. The antecedent specifies that the vectors A and B are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to F, G, and H.
|- A e. ~H   &   |- B e. ~H   &   |- F = (span` {A})   &   |- G = (span` {B})   &   |- H = (span` {(A +h B)})   =>   |- (-. (A e. G \/ B e. F) -> (H i^i (F vH G)) =/= ((H i^i F) vH (H i^i G)))
Theorem	spansncvi 11232	Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153.
|- A e. CH   &   |- B e. CH   &   |- C e. ~H   =>   |- ((A C. B /\ B C_ (A vH (span` {C}))) -> B = (A vH (span` {C})))
Theorem	spansncv 11233	Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153.
|- ((A e. CH /\ B e. CH /\ C e. ~H) -> ((A C. B /\ B C_ (A vH (span` {C}))) -> B = (A vH (span` {C}))))
Orthoarguesian laws 5OA and 3OA
Theorem	5oalem1 11234	Lemma for orthoarguesian law 5OA.
Theorem	5oalem2 11235	Lemma for orthoarguesian law 5OA.
Theorem	5oalem3 11236	Lemma for orthoarguesian law 5OA.
Theorem	5oalem4 11237	Lemma for orthoarguesian law 5OA.
Theorem	5oalem5 11238	Lemma for orthoarguesian law 5OA.
Theorem	5oalem6 11239	Lemma for orthoarguesian law 5OA.
Theorem	5oalem7 11240	Lemma for orthoarguesian law 5OA.
Theorem	5oai 11241	Orthoarguesian law 5OA. This 8-variable inference is called 5OA because it can be converted to a 5-variable equation (see Quantum Logic Explorer).
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- D e. CH   &   |- F e. CH   &   |- G e. CH   &   |- R e. CH   &   |- S e. CH   &   |- A C_ (_|_` B)   &   |- C C_ (_|_` D)   &   |- F C_ (_|_` G)   &   |- R C_ (_|_` S)   =>   |- (((A vH B) i^i (C vH D)) i^i ((F vH G) i^i (R vH S))) C_ (B vH (A i^i (C vH ((((A vH C) i^i (B vH D)) i^i (((A vH R) i^i (B vH S)) vH ((C vH R) i^i (D vH S)))) i^i ((((A vH F) i^i (B vH G)) i^i (((A vH R) i^i (B vH S)) vH ((F vH R) i^i (G vH S)))) vH (((C vH F) i^i (D vH G)) i^i (((C vH R) i^i (D vH S)) vH ((F vH R) i^i (G vH S)))))))))
Theorem	3oalem1 11242	Lemma for 3OA (weak) orthoarguesian law.
Theorem	3oalem2 11243	Lemma for 3OA (weak) orthoarguesian law.
Theorem	3oalem3 11244	Lemma for 3OA (weak) orthoarguesian law.
Theorem	3oalem4 11245	Lemma for 3OA (weak) orthoarguesian law.
Theorem	3oalem5 11246	Lemma for 3OA (weak) orthoarguesian law.
Theorem	3oalem6 11247	Lemma for 3OA (weak) orthoarguesian law.
Theorem	3oai 11248	3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- R = ((_|_` B) i^i (B vH A))   &   |- S = ((_|_` C) i^i (C vH A))   =>   |- ((B vH R) i^i (C vH S)) C_ (B vH (R i^i (S vH ((B vH C) i^i (R vH S)))))
Projectors (cont.)
Theorem	pjorthi 11249	Projection components on orthocomplemented subspaces are orthogonal.
|- A e. ~H   &   |- B e. ~H   =>   |- (H e. CH -> (((proj` H)` A) .ih ((proj` (_|_` H))` B)) = 0)
Theorem	pjch1 11250	Property of identity projection. Remark in [Beran] p. 111.
|- (A e. ~H -> ((proj` ~H)` A) = A)
Theorem	pjo 11251	The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111.
|- ((H e. CH /\ A e. ~H) -> ((proj` (_|_` H))` A) = (((proj` ~H)` A) -h ((proj` H)` A)))
Theorem	pjidmi 11252	A projection is idempotent. Property (ii) of [Beran] p. 109.
|- H e. CH   &   |- A e. ~H   =>   |- ((proj` H)` ((proj` H)` A)) = ((proj` H)` A)
Theorem	pjadjii 11253	A projection is self-adjoint. Property (i) of [Beran] p. 109.
|- H e. CH   &   |- A e. ~H   &   |- B e. ~H   =>   |- (((proj` H)` A) .ih B) = (A .ih ((proj` H)` B))
Theorem	pjcompi 11254	Component of a projection.
|- H e. CH   &   |- A e. ~H   &   |- B e. ~H   =>   |- ((A e. H /\ B e. (_|_` H)) -> ((proj` H)` (A +h B)) = A)
Theorem	pjaddii 11255	Projection of vector sum is sum of projections.
|- H e. CH   &   |- A e. ~H   &   |- B e. ~H   =>   |- ((proj` H)` (A +h B)) = (((proj` H)` A) +h ((proj` H)` B))
Theorem	pjinormii 11256	The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44.
|- H e. CH   &   |- A e. ~H   =>   |- (((proj` H)` A) .ih A) = ((normh` ((proj` H)` A))^2)
Theorem	pjmulii 11257	Projection of (scalar) product is product of projection.
|- H e. CH   &   |- A e. ~H   &   |- C e. CC   =>   |- ((proj` H)` (C .h A)) = (C .h ((proj` H)` A))
Theorem	pjsubii 11258	Projection of vector difference is difference of projections.
|- H e. CH   &   |- A e. ~H   &   |- B e. ~H   =>   |- ((proj` H)` (A -h B)) = (((proj` H)` A) -h ((proj` H)` B))
Theorem	pjsslem 11259	Lemma for subset relationships of projections.
Theorem	pjss2i 11260	Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112.
|- H e. CH   &   |- A e. ~H   &   |- G e. CH   =>   |- (H C_ G -> ((proj` H)` ((proj` G)` A)) = ((proj` H)` A))
Theorem	pjssmii 11261	Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112.
|- H e. CH   &   |- A e. ~H   &   |- G e. CH   =>   |- (H C_ G -> (((proj` G)` A) -h ((proj` H)` A)) = ((proj` (G i^i (_|_` H)))` A))
Theorem	pjssge0ii 11262	Theorem 4.5(iv)->(v) of [Beran] p. 112.
|- H e. CH   &   |- A e. ~H   &   |- G e. CH   =>   |- ((((proj` G)` A) -h ((proj` H)` A)) = ((proj` (G i^i (_|_` H)))` A) -> 0 <_ ((((proj` G)` A) -h ((proj` H)` A)) .ih A))
Theorem	pjdifnormii 11263	Theorem 4.5(v)<->(vi) of [Beran] p. 112.
|- H e. CH   &   |- A e. ~H   &   |- G e. CH   =>   |- (0 <_ ((((proj` G)` A) -h ((proj` H)` A)) .ih A) <-> (normh` ((proj` H)` A)) <_ (normh` ((proj` G)` A)))
Theorem	pjcji 11264	The projection on a subspace join is the sum of the projections.
|- H e. CH   &   |- A e. ~H   &   |- G e. CH   =>   |- (H C_ (_|_` G) -> ((proj` (H vH G))` A) = (((proj` H)` A) +h ((proj` G)` A)))
Theorem	pjadji 11265	A projection is self-adjoint. Property (i) of [Beran] p. 109.
|- H e. CH   =>   |- ((A e. ~H /\ B e. ~H) -> (((proj` H)` A) .ih B) = (A .ih ((proj` H)` B)))
Theorem	pjaddi 11266	Projection of vector sum is sum of projections.
|- H e. CH   =>   |- ((A e. ~H /\ B e. ~H) -> ((proj` H)` (A +h B)) = (((proj` H)` A) +h ((proj` H)` B)))
Theorem	pjinormi 11267	The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44.
|- H e. CH   =>   |- (A e. ~H -> (((proj` H)` A) .ih A) = ((normh` ((proj` H)` A))^2))
Theorem	pjsubi 11268	Projection of vector difference is difference of projections.
|- H e. CH   =>   |- ((A e. ~H /\ B e. ~H) -> ((proj` H)` (A -h B)) = (((proj` H)` A) -h ((proj` H)` B)))
Theorem	pjmuli 11269	Projection of scalar product is scalar product of projection.
|- H e. CH   =>   |- ((A e. CC /\ B e. ~H) -> ((proj` H)` (A .h B)) = (A .h ((proj` H)` B)))
Theorem	pjige0i 11270	The inner product of a projection and its argument is nonnegative.
|- H e. CH   =>   |- (A e. ~H -> 0 <_ (((proj` H)` A) .ih A))
Theorem	pjige0 11271	The inner product of a projection and its argument is nonnegative.
|- ((H e. CH /\ A e. ~H) -> 0 <_ (((proj` H)` A) .ih A))
Theorem	pjcjt2 11272	The projection on a subspace join is the sum of the projections.
|- ((H e. CH /\ G e. CH /\ A e. ~H) -> (H C_ (_|_` G) -> ((proj` (H vH G))` A) = (((proj` H)` A) +h ((proj` G)` A))))
Theorem	pj0i 11273	The projection of the zero vector.
|- H e. CH   =>   |- ((proj` H)` 0h) = 0h
Theorem	pjch 11274	Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111.
|- ((H e. CH /\ A e. ~H) -> (A e. H <-> ((proj` H)` A) = A))
Theorem	pjid 11275	The projection of a vector in the projection subspace is itself.
|- ((H e. CH /\ A e. H) -> ((proj` H)` A) = A)
Theorem	pjvec 11276	The set of vectors belonging to the subspace of a projection. Part of Theorem 26.2 of [Halmos] p. 44.
|- (H e. CH -> H = {x e. ~H | ((proj` H)` x) = x})
Theorem	pjocvec 11277	The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of [Halmos] p. 45.
|- (H e. CH -> (_|_` H) = {x e. ~H | ((proj` H)` x) = 0h})
Theorem	pjocini 11278	Membership of projection in orthocomplement of intersection.
|- G e. CH   &   |- H e. CH   =>   |- (A e. (_|_` (G i^i H)) -> ((proj` G)` A) e. (_|_` (G i^i H)))
Theorem	pjini 11279	Membership of projection in an intersection.
|- G e. CH   &   |- H e. CH   =>   |- (A e. (G i^i H) -> ((proj` G)` A) e. (G i^i H))
Theorem	pjjsi 11280	A sufficient condition for subspace join to be equal to subspace sum.
|- G e. CH   &   |- H e. SH   =>   |- (A.x e. (G vH H)((proj` (_|_` G))` x) e. H -> (G vH H) = (G +H H))
Theorem	pjfni 11281	Functionality of a projection.
|- H e. CH   =>   |- (proj` H) Fn ~H
Theorem	pjrni 11282	The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44.
|- H e. CH   =>   |- ran (proj` H) = H
Theorem	pjfoi 11283	A projection maps onto its subspace.
|- H e. CH   =>   |- (proj` H):~H-onto->H
Theorem	pjfi 11284	The mapping of a projection.
|- H e. CH   =>   |- (proj` H):~H-->~H
Theorem	pjvi 11285	The value of a projection in terms of components.
|- H e. CH   =>   |- ((A e. H /\ B e. (_|_` H)) -> ((proj` H)` (A +h B)) = A)
Theorem	pjfo 11286	A projection maps onto its subspace.
|- (H e. CH -> (proj` H):~H-onto->H)
Theorem	pjrn 11287	The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44.
|- (H e. CH -> ran (proj` H) = H)
Theorem	pjf 11288	The mapping of a projection.
|- (H e. CH -> (proj` H):~H-->~H)
Theorem	pjfn 11289	Functionality of a projection.
|- (H e. CH -> (proj` H) Fn ~H)
Theorem	pjsumi 11290	The projection on a subspace sum is the sum of the projections.
|- G e. CH   &   |- H e. CH   =>   |- (A e. ~H -> (G C_ (_|_` H) -> ((proj` (G +H H))` A) = (((proj` G)` A) +h ((proj` H)` A))))
Theorem	pj11i 11291	One-to-one correspondence of projection and subspace.
|- G e. CH   &   |- H e. CH   =>   |- ((proj` G) = (proj` H) <-> G = H)
Theorem	pjdsi 11292	Vector decomposition into sum of projections on orthogonal subspaces.
|- G e. CH   &   |- H e. CH   =>   |- ((A e. (G vH H) /\ G C_ (_|_` H)) -> A = (((proj` G)` A) +h ((proj` H)` A)))
Theorem	pjds3i 11293	Vector decomposition into sum of projections on orthogonal subspaces.
|- F e. CH   &   |- G e. CH   &   |- H e. CH   =>   |- (((A e. ((F vH G) vH H) /\ F C_ (_|_` G)) /\ (F C_ (_|_` H) /\ G C_ (_|_` H))) -> A = ((((proj` F)` A) +h ((proj` G)` A)) +h ((proj` H)` A)))
Theorem	pj11 11294	One-to-one correspondence of projection and subspace.
|- ((G e. CH /\ H e. CH) -> ((proj` G) = (proj` H) <-> G = H))
Theorem	pjmfn 11295	Functionality of the projection function.
|- proj Fn CH
Theorem	pjmf1 11296	The projector function maps one-to-one into the set of Hilbert space operators.
|- proj:CH-1-1->(~H ^m ~H)
Theorem	pjoi0 11297	The inner product of projections on orthogonal subspaces vanishes.
|- (((G e. CH /\ H e. CH /\ A e. ~H) /\ G C_ (_|_` H)) -> (((proj` G)` A) .ih ((proj` H)` A)) = 0)
Theorem	pjoi0i 11298	The inner product of projections on orthogonal subspaces vanishes.
|- G e. CH   &   |- H e. CH   &   |- A e. ~H   =>   |- (G C_ (_|_` H) -> (((proj` G)` A) .ih ((proj` H)` A)) = 0)
Theorem	pjopythi 11299	Pythagorean theorem for projections on orthogonal subspaces.
|- G e. CH   &   |- H e. CH   &   |- A e. ~H   =>   |- (G C_ (_|_` H) -> ((normh` (((proj` G)` A) +h ((proj` H)` A)))^2) = (((normh` ((proj` G)` A))^2) + ((normh` ((proj` H)` A))^2)))
Theorem	pjopyth 11300	Pythagorean theorem for projections on orthogonal subspaces.
|- ((H e. CH /\ G e. CH /\ A e. ~H) -> (H C_ (_|_` G) -> ((normh` (((proj` H)` A) +h ((proj` G)` A)))^2) = (((normh` ((proj` H)` A))^2) + ((normh` ((proj` G)` A))^2))))