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Theorem lsmelvalix 19659
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 2737 . . 3 (𝑋 + 𝑌) = (𝑋 + 𝑌)
2 rspceov 7480 . . 3 ((𝑋𝑇𝑌𝑈 ∧ (𝑋 + 𝑌) = (𝑋 + 𝑌)) → ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
31, 2mp3an3 1452 . 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 19658 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → ((𝑋 + 𝑌) ∈ (𝑇 𝑈) ↔ ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)))
87biimpar 477 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
93, 8sylan2 593 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  LSSumclsm 19652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-lsm 19654
This theorem is referenced by:  lsmub1x  19664  lsmub2x  19665  lsmelvali  19668  lsmsubm  19671  kercvrlsm  43095
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