Theorem lsmval 18063

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 17599	. . 3 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	adantr 481	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3		lsmval.v	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
4	3	subgss 17595	. . 3 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ 𝐵)
5	4	adantr 481	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ 𝐵)
6	3	subgss 17595	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
7	6	adantl 482	. 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 18054	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))
11	2, 5, 7, 10	syl3anc 1326	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ran crn 5115  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  +_gcplusg 15941  Grpcgrp 17422  SubGrpcsubg 17588  LSSumclsm 18049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-subg 17591  df-lsm 18051

This theorem is referenced by:  lsmidm  18077  lsmass  18083