Theorem lsmelvalix 19582

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 7417	. . 3 ⊢ ((𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈 ∧ (𝑋 + 𝑌) = (𝑋 + 𝑌)) → ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
3	1, 2	mp3an3 1453	. 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 19581	. . 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 594	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  LSSumclsm 19575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-lsm 19577

This theorem is referenced by:  lsmub1x  19587  lsmub2x  19588  lsmelvali  19591  lsmsubm  19594  kercvrlsm  43437