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Theorem lsmval 19585

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 19070	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		lsmval.v	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3	2	subgss 19066	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ 𝐵)
4	2	subgss 19066	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
5		lsmval.a	. . 3 ⊢ + = (+_g‘𝐺)
6		lsmval.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
7	2, 5, 6	lsmvalx 19576	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))
8	1, 3, 4, 7	syl2an3an 1424	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3917 ran crn 5642 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 Basecbs 17186 +_gcplusg 17227 Grpcgrp 18872 SubGrpcsubg 19059 LSSumclsm 19571

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-subg 19062 df-lsm 19573

This theorem is referenced by: lsmass 19606

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