lsmval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lsmval		Structured version Visualization version GIF version

Theorem lsmval 19681

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 19162	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		lsmval.v	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3	2	subgss 19158	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ 𝐵)
4	2	subgss 19158	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
5		lsmval.a	. . 3 ⊢ + = (+_g‘𝐺)
6		lsmval.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
7	2, 5, 6	lsmvalx 19672	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))
8	1, 3, 4, 7	syl2an3an 1421	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 1537 ∈ wcel 2106 ⊆ wss 3963 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 +_gcplusg 17298 Grpcgrp 18964 SubGrpcsubg 19151 LSSumclsm 19667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-subg 19154 df-lsm 19669

This theorem is referenced by: lsmass 19702

W3C validator