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Theorem lsmelvalix 18757
 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 2822 . . 3 (𝑋 + 𝑌) = (𝑋 + 𝑌)
2 rspceov 7187 . . 3 ((𝑋𝑇𝑌𝑈 ∧ (𝑋 + 𝑌) = (𝑋 + 𝑌)) → ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
31, 2mp3an3 1447 . 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 18756 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → ((𝑋 + 𝑌) ∈ (𝑇 𝑈) ↔ ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)))
87biimpar 481 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ ∃𝑥𝑇𝑦𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
93, 8sylan2 595 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∃wrex 3131   ⊆ wss 3908  ‘cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  LSSumclsm 18750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-lsm 18752 This theorem is referenced by:  lsmub1x  18762  lsmub2x  18763  lsmelvali  18766  lsmsubm  18769  kercvrlsm  39957
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