Theorem lsmless2x 19675

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 4004	. . . . 5 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
2	1	reximdv 3176	. . . 4 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
3	2	adantl 485	. . 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 1204	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝐺 ∈ 𝑉)
5		simpl2 1205	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑅 ⊆ 𝐵)
6		simpr 488	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈)
7		simpl3 1206	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑈 ⊆ 𝐵)
8	6, 7	sstrd 3944	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝐵)
9		lsmless2.v	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
10		eqid 2761	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
11		lsmless2.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
12	9, 10, 11	lsmelvalx 19670	. . . 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 1389	. . 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 19670	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑅 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧)))
15	14	adantr 484	. . 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 296	. 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑇) → 𝑥 ∈ (𝑅 ⊕ 𝑈)))
17	16	ssrdv 3940	1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃wrex 3085   ⊆ wss 3902  ‘cfv 6515  (class class class)co 7390  Basecbs 17235  +_gcplusg 17276  LSSumclsm 19664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-lsm 19666

This theorem is referenced by:  lsmless2  19691  lsmssspx  21142