Theorem lsmval 17886

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	⊢ + = (+g‘𝐺)
lsmval.p	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
lsmval	⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (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

Step	Hyp	Ref	Expression
1		subgrcl 17422	. . 3 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	adantr 480	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3		lsmval.v	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgss 17418	. . 3 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ 𝐵)
5	4	adantr 480	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ 𝐵)
6	3	subgss 17418	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
7	6	adantl 481	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ 𝐵)
8		lsmval.a	. . 3 ⊢ + = (+g‘𝐺)
9		lsmval.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
10	3, 8, 9	lsmvalx 17877	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))
11	2, 5, 7, 10	syl3anc 1318	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +_gcplusg 15768  Grpcgrp 17245  SubGrpcsubg 17411  LSSumclsm 17872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-subg 17414  df-lsm 17874

This theorem is referenced by:  lsmidm  17900  lsmass  17906