Theorem lsmless2x 19604

Description: Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmless2.v	⊢ 𝐵 = (Base‘𝐺)
lsmless2.s	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
lsmless2x	⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈))

Proof of Theorem lsmless2x

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrexv 4042	. . . . 5 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
2	1	reximdv 3160	. . . 4 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
3	2	adantl 480	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
4		simpl1 1188	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝐺 ∈ 𝑉)
5		simpl2 1189	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑅 ⊆ 𝐵)
6		simpr 483	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈)
7		simpl3 1190	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑈 ⊆ 𝐵)
8	6, 7	sstrd 3983	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝐵)
9		lsmless2.v	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
10		eqid 2725	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
11		lsmless2.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
12	9, 10, 11	lsmelvalx 19599	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑥 ∈ (𝑅 ⊕ 𝑇) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧)))
13	4, 5, 8, 12	syl3anc 1368	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑇) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧)))
14	9, 10, 11	lsmelvalx 19599	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑅 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
15	14	adantr 479	. . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
16	3, 13, 15	3imtr4d 293	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑇) → 𝑥 ∈ (𝑅 ⊕ 𝑈)))
17	16	ssrdv 3978	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∃wrex 3060   ⊆ wss 3939  ‘cfv 6543  (class class class)co 7416  Basecbs 17179  +_gcplusg 17232  LSSumclsm 19593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-lsm 19595

This theorem is referenced by:  lsmless2  19620  lsmssspx  20977