Theorem lsmelvali 19778

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
lsmelvali	⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))

Proof of Theorem lsmelvali

Step	Hyp	Ref	Expression
1		subgrcl 19255	. . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	adantr 486	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3		eqid 2760	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	3	subgss 19251	. . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
5	4	adantr 486	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (Base‘𝐺))
6	3	subgss 19251	. . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7	6	adantl 487	. . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (Base‘𝐺))
8	2, 5, 7	3jca 1146	. 2 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)))
9		lsmelval.a	. . 3 ⊢ + = (+g‘𝐺)
10		lsmelval.p	. . 3 ⊢ ⊕ = (LSSum‘𝐺)
11	3, 9, 10	lsmelvalix 19769	. 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))
12	8, 11	sylan 592	1 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  +_gcplusg 17343  Grpcgrp 19058  SubGrpcsubg 19244  LSSumclsm 19762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-subg 19247  df-lsm 19764

This theorem is used by:  lsmsubg  19782  lsmmod  19803  lsmdisj2  19810  lsmhash  19833  ablfacrp  20196  lsmcl  21268  lsmelval2  21270  lsppreli  21275  lspprabs  21280  lspabs3  21309  pjthlem2  25667  lkrlsp  39976  dia2dimlem5  41942  mapdindp0  42593  hdmaprnlem3eN  42732