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Theorem lsmelvalx 19599
Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19608. (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 19598 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)))
54eleq2d 2811 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ 𝑋 ∈ ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧))))
6 eqid 2725 . . 3 (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)) = (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧))
7 ovex 7449 . . 3 (𝑦 + 𝑧) ∈ V
86, 7elrnmpo 7554 . 2 (𝑋 ∈ ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧))
95, 8bitrdi 286 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  wrex 3060  wss 3939  ran crn 5673  cfv 6543  (class class class)co 7416  cmpo 7418  Basecbs 17179  +gcplusg 17232  LSSumclsm 19593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-lsm 19595
This theorem is referenced by:  lsmelvalix  19600  lsmless1x  19603  lsmless2x  19604  lsmelval  19608  lsmsubm  19612  lsmass  19628  lsmcomx  19815  lsmcss  21628  elgrplsmsn  33148  elringlsm  33151  lsmssass  33160  grplsm0l  33161  grplsmid  33162  mxidlprm  33232  dimkerim  33382
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