MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmless2x Structured version   Visualization version   GIF version

Theorem lsmless2x 19664
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 4052 . . . . 5 (𝑇𝑈 → (∃𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
21reximdv 3169 . . . 4 (𝑇𝑈 → (∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
32adantl 481 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
4 simpl1 1191 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝐺𝑉)
5 simpl2 1192 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑅𝐵)
6 simpr 484 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝑈)
7 simpl3 1193 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑈𝐵)
86, 7sstrd 3993 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝐵)
9 lsmless2.v . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2736 . . . . 5 (+g𝐺) = (+g𝐺)
11 lsmless2.s . . . . 5 = (LSSum‘𝐺)
129, 10, 11lsmelvalx 19659 . . . 4 ((𝐺𝑉𝑅𝐵𝑇𝐵) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
134, 5, 8, 12syl3anc 1372 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
149, 10, 11lsmelvalx 19659 . . . 4 ((𝐺𝑉𝑅𝐵𝑈𝐵) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
1514adantr 480 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
163, 13, 153imtr4d 294 . 2 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑇) → 𝑥 ∈ (𝑅 𝑈)))
1716ssrdv 3988 1 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wrex 3069  wss 3950  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  LSSumclsm 19653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-lsm 19655
This theorem is referenced by:  lsmless2  19680  lsmssspx  21088
  Copyright terms: Public domain W3C validator