Theorem lsmless2x 19772

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 4001	. . . . 5 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
2	1	reximdv 3177	. . . 4 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
3	2	adantl 487	. . 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 1210	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝐺 ∈ 𝑉)
5		simpl2 1211	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑅 ⊆ 𝐵)
6		simpr 490	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈)
7		simpl3 1212	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑈 ⊆ 𝐵)
8	6, 7	sstrd 3941	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝐵)
9		lsmless2.v	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
10		eqid 2760	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
11		lsmless2.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
12	9, 10, 11	lsmelvalx 19767	. . . 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 1398	. . 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 19767	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑅 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
15	14	adantr 486	. . 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 297	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑇) → 𝑥 ∈ (𝑅 ⊕ 𝑈)))
17	16	ssrdv 3937	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3899  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  +_gcplusg 17342  LSSumclsm 19761

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-lsm 19763

This theorem is used by:  lsmless2  19788  lsmssspx  21272