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Theorem lsmval 18702
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 18222 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2 lsmval.v . . 3 𝐵 = (Base‘𝐺)
32subgss 18218 . 2 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇𝐵)
42subgss 18218 . 2 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
5 lsmval.a . . 3 + = (+g𝐺)
6 lsmval.p . . 3 = (LSSum‘𝐺)
72, 5, 6lsmvalx 18693 . 2 ((𝐺 ∈ Grp ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
81, 3, 4, 7syl2an3an 1414 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 396   = wceq 1528  wcel 2105  wss 3933  ran crn 5549  cfv 6348  (class class class)co 7145  cmpo 7147  Basecbs 16471  +gcplusg 16553  Grpcgrp 18041  SubGrpcsubg 18211  LSSumclsm 18688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-subg 18214  df-lsm 18690
This theorem is referenced by:  lsmidmOLD  18718  lsmass  18724
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