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Theorem lsmless1x 18761
Description: Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmless2.v 𝐵 = (Base‘𝐺)
lsmless2.s = (LSSum‘𝐺)
Assertion
Ref Expression
lsmless1x (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑅 𝑈) ⊆ (𝑇 𝑈))

Proof of Theorem lsmless1x
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrexv 3982 . . . 4 (𝑅𝑇 → (∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
21adantl 485 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
3 simpl1 1188 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝐺𝑉)
4 simpr 488 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑅𝑇)
5 simpl2 1189 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑇𝐵)
64, 5sstrd 3925 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑅𝐵)
7 simpl3 1190 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑈𝐵)
8 lsmless2.v . . . . 5 𝐵 = (Base‘𝐺)
9 eqid 2798 . . . . 5 (+g𝐺) = (+g𝐺)
10 lsmless2.s . . . . 5 = (LSSum‘𝐺)
118, 9, 10lsmelvalx 18757 . . . 4 ((𝐺𝑉𝑅𝐵𝑈𝐵) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
123, 6, 7, 11syl3anc 1368 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
138, 9, 10lsmelvalx 18757 . . . 4 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
1413adantr 484 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
152, 12, 143imtr4d 297 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑥 ∈ (𝑅 𝑈) → 𝑥 ∈ (𝑇 𝑈)))
1615ssrdv 3921 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑅 𝑈) ⊆ (𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wrex 3107  wss 3881  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  LSSumclsm 18751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-lsm 18753
This theorem is referenced by:  lsmless1  18777  lsmless12  18779  lsmssspx  19853
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