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

Theorem lsmelvalix 18440
Description: Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmfval.v 𝐵 = (Base‘𝐺)
lsmfval.a + = (+g𝐺)
lsmfval.s = (LSSum‘𝐺)
Assertion
Ref Expression
lsmelvalix (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))

Proof of Theorem lsmelvalix
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2777 . . 3 (𝑋 + 𝑌) = (𝑋 + 𝑌)
2 rspceov 6968 . . 3 ((𝑋𝑇𝑌𝑈 ∧ (𝑋 + 𝑌) = (𝑋 + 𝑌)) → ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
31, 2mp3an3 1523 . 2 ((𝑋𝑇𝑌𝑈) → ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
4 lsmfval.v . . . 4 𝐵 = (Base‘𝐺)
5 lsmfval.a . . . 4 + = (+g𝐺)
6 lsmfval.s . . . 4 = (LSSum‘𝐺)
74, 5, 6lsmelvalx 18439 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → ((𝑋 + 𝑌) ∈ (𝑇 𝑈) ↔ ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)))
87biimpar 471 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
93, 8sylan2 586 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  wrex 3090  wss 3791  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  LSSumclsm 18433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-lsm 18435
This theorem is referenced by:  lsmub1x  18445  lsmub2x  18446  lsmelvali  18449  lsmsubm  18452  kercvrlsm  38605
  Copyright terms: Public domain W3C validator