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Theorem lsmval 18283
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 17820 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 472 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3 lsmval.v . . . 4 𝐵 = (Base‘𝐺)
43subgss 17816 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇𝐵)
54adantr 472 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇𝐵)
63subgss 17816 . . 3 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
76adantl 473 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈𝐵)
8 lsmval.a . . 3 + = (+g𝐺)
9 lsmval.p . . 3 = (LSSum‘𝐺)
103, 8, 9lsmvalx 18274 . 2 ((𝐺 ∈ Grp ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
112, 5, 7, 10syl3anc 1477 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 383   = wceq 1632  wcel 2139  wss 3715  ran crn 5267  cfv 6049  (class class class)co 6814  cmpt2 6816  Basecbs 16079  +gcplusg 16163  Grpcgrp 17643  SubGrpcsubg 17809  LSSumclsm 18269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-subg 17812  df-lsm 18271
This theorem is referenced by:  lsmidm  18297  lsmass  18303
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