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Theorem lsmelvalx 19560
Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19569. (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 19559 . . 3 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)))
54eleq2d 2813 . 2 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ 𝑋 ∈ ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧))))
6 eqid 2726 . . 3 (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)) = (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧))
7 ovex 7438 . . 3 (𝑦 + 𝑧) ∈ V
86, 7elrnmpo 7541 . 2 (𝑋 ∈ ran (𝑦𝑇, 𝑧𝑈 ↦ (𝑦 + 𝑧)) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧))
95, 8bitrdi 287 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 3064  wss 3943  ran crn 5670  cfv 6537  (class class class)co 7405  cmpo 7407  Basecbs 17153  +gcplusg 17206  LSSumclsm 19554
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-lsm 19556
This theorem is referenced by:  lsmelvalix  19561  lsmless1x  19564  lsmless2x  19565  lsmelval  19569  lsmsubm  19573  lsmass  19589  lsmcomx  19776  lsmcss  21585  elgrplsmsn  33006  elringlsm  33009  lsmssass  33018  grplsm0l  33019  grplsmid  33020  mxidlprm  33092  dimkerim  33230
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