Theorem lsmelvalix 19835

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 2761	. . 3 ⊢ (𝑋 + 𝑌) = (𝑋 + 𝑌)
2		rspceov 7461	. . 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 19834	. . 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 2145  ∃wrex 3087   ⊆ wss 3899  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  +_gcplusg 17408  LSSumclsm 19828

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-lsm 19830

This theorem is used by:  lsmub1x  19840  lsmub2x  19841  lsmelvali  19844  lsmsubm  19847  kercvrlsm  44043