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Theorem lsmelvalx 18756
Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 18765. (Contributed by NM, 28-Jan-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
lsmelvalx ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
Distinct variable groups:   𝑦,𝑧, +   𝑦,𝐵,𝑧   𝑦,𝑇,𝑧   𝑦,𝑋,𝑧   𝑦,𝐺,𝑧   𝑦,𝑈,𝑧
Allowed substitution hints:   (𝑦,𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem lsmelvalx
StepHypRef Expression
1 lsmfval.v . . . 4 𝐵 = (Base‘𝐺)
2 lsmfval.a . . . 4 + = (+g𝐺)
3 lsmfval.s . . . 4 = (LSSum‘𝐺)
41, 2, 3lsmvalx 18755 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)))
54eleq2d 2899 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ 𝑋 ∈ ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧))))
6 eqid 2822 . . 3 (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)) = (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧))
7 ovex 7173 . . 3 (𝑦 + 𝑧) ∈ V
86, 7elrnmpo 7271 . 2 (𝑋 ∈ ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧))
95, 8syl6bb 290 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2114  wrex 3131  wss 3908  ran crn 5533  cfv 6334  (class class class)co 7140  cmpo 7142  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:  lsmelvalix  18757  lsmless1x  18760  lsmless2x  18761  lsmelval  18765  lsmsubm  18769  lsmass  18786  lsmcomx  18967  lsmcss  20379  elgrplsmsn  30979  elringlsm  30981  lsmssass  30990  mxidlprm  31019  dimkerim  31080
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