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Theorem shocel 28697
 Description: Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shocel (𝐻S → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
Distinct variable groups:   𝑥,𝐻   𝑥,𝐴

Proof of Theorem shocel
StepHypRef Expression
1 shss 28623 . 2 (𝐻S𝐻 ⊆ ℋ)
2 ocel 28696 . 2 (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
31, 2syl 17 1 (𝐻S → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3118   ⊆ wss 3799  ‘cfv 6124  (class class class)co 6906  0cc0 10253   ℋchba 28332   ·ih csp 28335   Sℋ csh 28341  ⊥cort 28343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-hilex 28412 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fv 6132  df-ov 6909  df-sh 28620  df-oc 28665 This theorem is referenced by:  ocin  28711  choc0  28741  choc1  28742  pjhthlem2  28807  pjclem4  29614  pj3si  29622
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