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 19666

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 19149	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		lsmval.v	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3	2	subgss 19145	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ 𝐵)
4	2	subgss 19145	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
5		lsmval.a	. . 3 ⊢ + = (+_g‘𝐺)
6		lsmval.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
7	2, 5, 6	lsmvalx 19657	. 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 2108 ⊆ wss 3951 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +_gcplusg 17297 Grpcgrp 18951 SubGrpcsubg 19138 LSSumclsm 19652

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-subg 19141 df-lsm 19654

This theorem is referenced by: lsmass 19687

W3C validator