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Theorem lsmless2x 19678
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 4065 . . . . 5 (𝑇𝑈 → (∃𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
21reximdv 3168 . . . 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 1190 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝐺𝑉)
5 simpl2 1191 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑅𝐵)
6 simpr 484 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝑈)
7 simpl3 1192 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑈𝐵)
86, 7sstrd 4006 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝐵)
9 lsmless2.v . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2735 . . . . 5 (+g𝐺) = (+g𝐺)
11 lsmless2.s . . . . 5 = (LSSum‘𝐺)
129, 10, 11lsmelvalx 19673 . . . 4 ((𝐺𝑉𝑅𝐵𝑇𝐵) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
134, 5, 8, 12syl3anc 1370 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
149, 10, 11lsmelvalx 19673 . . . 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 4001 1 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  LSSumclsm 19667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-lsm 19669
This theorem is referenced by:  lsmless2  19694  lsmssspx  21105
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