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

Theorem restabs 23188
Description: Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restabs ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))

Proof of Theorem restabs
StepHypRef Expression
1 simp1 1135 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝐽𝑉)
2 simp3 1137 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑇𝑊)
3 ssexg 5328 . . . 4 ((𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
433adant1 1129 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
5 restco 23187 . . 3 ((𝐽𝑉𝑇𝑊𝑆 ∈ V) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
61, 2, 4, 5syl3anc 1370 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
7 simp2 1136 . . . 4 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆𝑇)
8 sseqin2 4230 . . . 4 (𝑆𝑇 ↔ (𝑇𝑆) = 𝑆)
97, 8sylib 218 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝑇𝑆) = 𝑆)
109oveq2d 7446 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝐽t (𝑇𝑆)) = (𝐽t 𝑆))
116, 10eqtrd 2774 1 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  wss 3962  (class class class)co 7430  t crest 17466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-rest 17468
This theorem is referenced by:  restcnrm  23385  fiuncmp  23427  subislly  23504  restnlly  23505  islly2  23507  llyrest  23508  nllyrest  23509  llyidm  23511  nllyidm  23512  cldllycmp  23518  txkgen  23675  rerest  24839  xrrest  24842  cnmpopc  24968  cnheiborlem  24999  pcoass  25070  limcres  25935  perfdvf  25952  dvreslem  25958  dvres2lem  25959  dvaddbr  25988  dvmulbr  25989  dvmulbrOLD  25990  dvcnvrelem2  26071  psercn  26484  abelth  26499  cxpcn2  26803  cxpcn3  26805  lmlimxrge0  33908  pnfneige0  33911  cvmsss2  35258  cvmliftlem8  35276  cvmliftlem10  35278  cvmlift2lem9  35295  ivthALT  36317  limcresiooub  45597  limcresioolb  45598  cncfuni  45841  cncfiooicclem1  45848  itgsubsticclem  45930  dirkercncflem4  46061  fourierdlem32  46094  fourierdlem33  46095  fourierdlem62  46123  fouriersw  46186  smfco  46757
  Copyright terms: Public domain W3C validator