Theorem lsmelvalix 19742

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 2766	. . 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 1479	. 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 19741	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → ((𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)))
8	7	biimpar 483	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 (𝑋 + 𝑌) = (𝑥 + 𝑦)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))
9	3, 8	sylan2 605	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3092   ⊆ wss 3908  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  LSSumclsm 19735

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-lsm 19737

This theorem is used by:  lsmub1x  19747  lsmub2x  19748  lsmelvali  19751  lsmsubm  19754  kercvrlsm  43851