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Theorem lsmless1x 17980
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 3646 . . . 4 (𝑅𝑇 → (∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
21adantl 482 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
3 simpl1 1062 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝐺𝑉)
4 simpr 477 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑅𝑇)
5 simpl2 1063 . . . . 5 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑇𝐵)
64, 5sstrd 3593 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑅𝐵)
7 simpl3 1064 . . . 4 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → 𝑈𝐵)
8 lsmless2.v . . . . 5 𝐵 = (Base‘𝐺)
9 eqid 2621 . . . . 5 (+g𝐺) = (+g𝐺)
10 lsmless2.s . . . . 5 = (LSSum‘𝐺)
118, 9, 10lsmelvalx 17976 . . . 4 ((𝐺𝑉𝑅𝐵𝑈𝐵) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
123, 6, 7, 11syl3anc 1323 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
138, 9, 10lsmelvalx 17976 . . . 4 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
1413adantr 481 . . 3 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
152, 12, 143imtr4d 283 . 2 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑥 ∈ (𝑅 𝑈) → 𝑥 ∈ (𝑇 𝑈)))
1615ssrdv 3589 1 (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑅 𝑈) ⊆ (𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  wss 3555  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  LSSumclsm 17970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-lsm 17972
This theorem is referenced by:  lsmless1  17995  lsmless12  17997  lsmssspx  19007
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