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Theorem lsmval 19666
Description: Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmval.v 𝐵 = (Base‘𝐺)
lsmval.a + = (+g𝐺)
lsmval.p = (LSSum‘𝐺)
Assertion
Ref Expression
lsmval ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
Distinct variable groups:   𝑥,𝑦, +   𝑥,𝑇,𝑦   𝑥,𝑈,𝑦   𝑥,𝐵,𝑦   𝑥,𝐺,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem lsmval
StepHypRef Expression
1 subgrcl 19149 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2 lsmval.v . . 3 𝐵 = (Base‘𝐺)
32subgss 19145 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇𝐵)
42subgss 19145 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
5 lsmval.a . . 3 + = (+g𝐺)
6 lsmval.p . . 3 = (LSSum‘𝐺)
72, 5, 6lsmvalx 19657 . 2 ((𝐺 ∈ Grp ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
81, 3, 4, 7syl2an3an 1424 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  SubGrpcsubg 19138  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-subg 19141  df-lsm 19654
This theorem is referenced by:  lsmass  19687
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