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

Theorem restabs 21017
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 1081 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝐽𝑉)
2 simp3 1083 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑇𝑊)
3 ssexg 4837 . . . 4 ((𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
433adant1 1099 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆 ∈ V)
5 restco 21016 . . 3 ((𝐽𝑉𝑇𝑊𝑆 ∈ V) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
61, 2, 4, 5syl3anc 1366 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t (𝑇𝑆)))
7 simp2 1082 . . . 4 ((𝐽𝑉𝑆𝑇𝑇𝑊) → 𝑆𝑇)
8 sseqin2 3850 . . . 4 (𝑆𝑇 ↔ (𝑇𝑆) = 𝑆)
97, 8sylib 208 . . 3 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝑇𝑆) = 𝑆)
109oveq2d 6706 . 2 ((𝐽𝑉𝑆𝑇𝑇𝑊) → (𝐽t (𝑇𝑆)) = (𝐽t 𝑆))
116, 10eqtrd 2685 1 ((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  wss 3607  (class class class)co 6690  t crest 16128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-rest 16130
This theorem is referenced by:  restcnrm  21214  fiuncmp  21255  subislly  21332  restnlly  21333  islly2  21335  llyrest  21336  nllyrest  21337  llyidm  21339  nllyidm  21340  cldllycmp  21346  txkgen  21503  rerest  22654  xrrest  22657  cnmpt2pc  22774  cnheiborlem  22800  pcoass  22870  limcres  23695  perfdvf  23712  dvreslem  23718  dvres2lem  23719  dvaddbr  23746  dvmulbr  23747  dvcnvrelem2  23826  psercn  24225  abelth  24240  cxpcn2  24532  cxpcn3  24534  lmlimxrge0  30122  pnfneige0  30125  cvmsss2  31382  cvmliftlem8  31400  cvmliftlem10  31402  cvmlift2lem9  31419  ivthALT  32455  limcresiooub  40192  limcresioolb  40193  cncfuni  40417  cncfiooicclem1  40424  itgsubsticclem  40509  dirkercncflem4  40641  fourierdlem32  40674  fourierdlem33  40675  fourierdlem62  40703  fouriersw  40766  smfco  41330
  Copyright terms: Public domain W3C validator