MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ocv2ss Structured version   Visualization version   GIF version

Theorem ocv2ss 21555
Description: Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypothesis
Ref Expression
ocv2ss.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
ocv2ss (𝑇 βŠ† 𝑆 β†’ ( βŠ₯ β€˜π‘†) βŠ† ( βŠ₯ β€˜π‘‡))

Proof of Theorem ocv2ss
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr2 3982 . . . 4 (𝑇 βŠ† 𝑆 β†’ (𝑆 βŠ† (Baseβ€˜π‘Š) β†’ 𝑇 βŠ† (Baseβ€˜π‘Š)))
2 idd 24 . . . 4 (𝑇 βŠ† 𝑆 β†’ (π‘₯ ∈ (Baseβ€˜π‘Š) β†’ π‘₯ ∈ (Baseβ€˜π‘Š)))
3 ssralv 4043 . . . 4 (𝑇 βŠ† 𝑆 β†’ (βˆ€π‘¦ ∈ 𝑆 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) β†’ βˆ€π‘¦ ∈ 𝑇 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
41, 2, 33anim123d 1439 . . 3 (𝑇 βŠ† 𝑆 β†’ ((𝑆 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝑆 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) β†’ (𝑇 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝑇 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
5 eqid 2724 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
6 eqid 2724 . . . 4 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
7 eqid 2724 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
8 eqid 2724 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
9 ocv2ss.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
105, 6, 7, 8, 9elocv 21550 . . 3 (π‘₯ ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝑆 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
115, 6, 7, 8, 9elocv 21550 . . 3 (π‘₯ ∈ ( βŠ₯ β€˜π‘‡) ↔ (𝑇 βŠ† (Baseβ€˜π‘Š) ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝑇 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
124, 10, 113imtr4g 296 . 2 (𝑇 βŠ† 𝑆 β†’ (π‘₯ ∈ ( βŠ₯ β€˜π‘†) β†’ π‘₯ ∈ ( βŠ₯ β€˜π‘‡)))
1312ssrdv 3981 1 (𝑇 βŠ† 𝑆 β†’ ( βŠ₯ β€˜π‘†) βŠ† ( βŠ₯ β€˜π‘‡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3941  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  Scalarcsca 17205  Β·π‘–cip 17207  0gc0g 17390  ocvcocv 21542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-ocv 21545
This theorem is referenced by:  ocvsscon  21557  ocvlsp  21558  ocvcss  21569  cssmre  21575  mrccss  21576  clsocv  25122
  Copyright terms: Public domain W3C validator