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Theorem lsmless2x 19034
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 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))

Proof of Theorem lsmless2x
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrexv 3968 . . . . 5 (𝑇𝑈 → (∃𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
21reximdv 3192 . . . 4 (𝑇𝑈 → (∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
32adantl 485 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
4 simpl1 1193 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝐺𝑉)
5 simpl2 1194 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑅𝐵)
6 simpr 488 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝑈)
7 simpl3 1195 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑈𝐵)
86, 7sstrd 3911 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝐵)
9 lsmless2.v . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2737 . . . . 5 (+g𝐺) = (+g𝐺)
11 lsmless2.s . . . . 5 = (LSSum‘𝐺)
129, 10, 11lsmelvalx 19029 . . . 4 ((𝐺𝑉𝑅𝐵𝑇𝐵) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
134, 5, 8, 12syl3anc 1373 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
149, 10, 11lsmelvalx 19029 . . . 4 ((𝐺𝑉𝑅𝐵𝑈𝐵) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
1514adantr 484 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
163, 13, 153imtr4d 297 . 2 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑇) → 𝑥 ∈ (𝑅 𝑈)))
1716ssrdv 3907 1 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wrex 3062  wss 3866  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  LSSumclsm 19023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-lsm 19025
This theorem is referenced by:  lsmless2  19050  lsmssspx  20125
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