Theorem lsmelval 19250

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 18756	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		eqid 2740	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	2	subgss 18752	. 2 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
4	2	subgss 18752	. 2 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5		lsmelval.a	. . 3 ⊢ + = (+g‘𝐺)
6		lsmelval.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
7	2, 5, 6	lsmelvalx 19241	. 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 1421	1 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∃wrex 3067   ⊆ wss 3892  ‘cfv 6431  (class class class)co 7269  Basecbs 16908  +_gcplusg 16958  Grpcgrp 18573  SubGrpcsubg 18745  LSSumclsm 19235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823  df-subg 18748  df-lsm 19237

This theorem is referenced by:  lsmelvalm  19252  lsmsubg  19255  lsmcom2  19256  lsmmod  19277  lsmdisj2  19284  pj1eu  19298  lsmcl  20341  lsmspsn  20342  lsmelval2  20343  lsmcv  20399  lindsunlem  31699  lsmsat  37016  lshpsmreu  37117  dvhopellsm  39125  diblsmopel  39179  cdlemn11c  39217  dihord11c  39232  hdmapglem7a  39935