Theorem lsmelvalix 19659

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

Hypotheses

Ref	Expression
lsmfval.v	⊢ 𝐵 = (Base‘𝐺)
lsmfval.a	⊢ + = (+g‘𝐺)
lsmfval.s	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
lsmelvalix	⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))

Proof of Theorem lsmelvalix

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (𝑋 + 𝑌) = (𝑋 + 𝑌)
2		rspceov 7480	. . 3 ⊢ ((𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈 ∧ (𝑋 + 𝑌) = (𝑋 + 𝑌)) → ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
3	1, 2	mp3an3 1452	. 2 ⊢ ((𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈) → ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
4		lsmfval.v	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
5		lsmfval.a	. . . 4 ⊢ + = (+g‘𝐺)
6		lsmfval.s	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
7	4, 5, 6	lsmelvalx 19658	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → ((𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)))
8	7	biimpar 477	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))
9	3, 8	sylan2 593	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  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-lsm 19654

This theorem is referenced by:  lsmub1x  19664  lsmub2x  19665  lsmelvali  19668  lsmsubm  19671  kercvrlsm  43095