Theorem lsmelvalx 18406

Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 18415. (Contributed by NM, 28-Jan-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
lsmelvalx	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))

Distinct variable groups:   𝑦,𝑧, +   𝑦,𝐵,𝑧   𝑦,𝑇,𝑧   𝑦,𝑋,𝑧   𝑦,𝐺,𝑧   𝑦,𝑈,𝑧

Allowed substitution hints:   ⊕ (𝑦,𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem lsmelvalx

Step	Hyp	Ref	Expression
1		lsmfval.v	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		lsmfval.a	. . . 4 ⊢ + = (+g‘𝐺)
3		lsmfval.s	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
4	1, 2, 3	lsmvalx 18405	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)))
5	4	eleq2d 2892	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑋 ∈ ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧))))
6		eqid 2825	. . 3 ⊢ (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) = (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧))
7		ovex 6937	. . 3 ⊢ (𝑦 + 𝑧) ∈ V
8	6, 7	elrnmpt2 7033	. 2 ⊢ (𝑋 ∈ ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))
9	5, 8	syl6bb 279	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∃wrex 3118   ⊆ wss 3798  ran crn 5343  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907  Basecbs 16222  +_gcplusg 16305  LSSumclsm 18400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-lsm 18402

This theorem is referenced by:  lsmelvalix  18407  lsmless1x  18410  lsmless2x  18411  lsmelval  18415  lsmsubm  18419  lsmass  18434  lsmcomx  18612  lsmcss  20399