Theorem lsmelvalix 19639

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 2726	. . 3 ⊢ (𝑋 + 𝑌) = (𝑋 + 𝑌)
2		rspceov 7472	. . 3 ⊢ ((𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈 ∧ (𝑋 + 𝑌) = (𝑋 + 𝑌)) → ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦))
3	1, 2	mp3an3 1447	. 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 19638	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → ((𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)))
8	7	biimpar 476	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))
9	3, 8	sylan2 591	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∃wrex 3060   ⊆ wss 3947  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  +_gcplusg 17266  LSSumclsm 19632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-lsm 19634

This theorem is referenced by:  lsmub1x  19644  lsmub2x  19645  lsmelvali  19648  lsmsubm  19651  kercvrlsm  42744