Theorem lsmelval 19616

Description: Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmelval.a	⊢ + = (+g‘𝐺)
lsmelval.p	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
lsmelval	⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))

Distinct variable groups:   𝑦,𝑧, +   𝑦,𝑇,𝑧   𝑦,𝑈,𝑧   𝑦,𝐺,𝑧   𝑦,𝑋,𝑧

Allowed substitution hints:   ⊕ (𝑦,𝑧)

Proof of Theorem lsmelval

Step	Hyp	Ref	Expression
1		subgrcl 19094	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqid 2725	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	2	subgss 19090	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
4	2	subgss 19090	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5		lsmelval.a	. . 3 ⊢ + = (+g‘𝐺)
6		lsmelval.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
7	2, 5, 6	lsmelvalx 19607	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))
8	1, 3, 4, 7	syl2an3an 1419	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059   ⊆ wss 3944  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  +_gcplusg 17236  Grpcgrp 18898  SubGrpcsubg 19083  LSSumclsm 19601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-subg 19086  df-lsm 19603

This theorem is referenced by:  lsmelvalm  19618  lsmsubg  19621  lsmcom2  19622  lsmmod  19642  lsmdisj2  19649  pj1eu  19663  lsmcl  20980  lsmspsn  20981  lsmelval2  20982  lsmcv  21041  lindsunlem  33450  lsmsat  38607  lshpsmreu  38708  dvhopellsm  40717  diblsmopel  40771  cdlemn11c  40809  dihord11c  40824  hdmapglem7a  41527