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Theorem lsmless2x 18500
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 3955 . . . . 5 (𝑇𝑈 → (∃𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
21reximdv 3236 . . . 4 (𝑇𝑈 → (∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
32adantl 482 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧) → ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
4 simpl1 1184 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝐺𝑉)
5 simpl2 1185 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑅𝐵)
6 simpr 485 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝑈)
7 simpl3 1186 . . . . 5 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑈𝐵)
86, 7sstrd 3899 . . . 4 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → 𝑇𝐵)
9 lsmless2.v . . . . 5 𝐵 = (Base‘𝐺)
10 eqid 2795 . . . . 5 (+g𝐺) = (+g𝐺)
11 lsmless2.s . . . . 5 = (LSSum‘𝐺)
129, 10, 11lsmelvalx 18495 . . . 4 ((𝐺𝑉𝑅𝐵𝑇𝐵) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
134, 5, 8, 12syl3anc 1364 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑇) ↔ ∃𝑦𝑅𝑧𝑇 𝑥 = (𝑦(+g𝐺)𝑧)))
149, 10, 11lsmelvalx 18495 . . . 4 ((𝐺𝑉𝑅𝐵𝑈𝐵) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
1514adantr 481 . . 3 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑈) ↔ ∃𝑦𝑅𝑧𝑈 𝑥 = (𝑦(+g𝐺)𝑧)))
163, 13, 153imtr4d 295 . 2 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑥 ∈ (𝑅 𝑇) → 𝑥 ∈ (𝑅 𝑈)))
1716ssrdv 3895 1 (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wrex 3106  wss 3859  cfv 6225  (class class class)co 7016  Basecbs 16312  +gcplusg 16394  LSSumclsm 18489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-lsm 18491
This theorem is referenced by:  lsmless2  18515  lsmssspx  19550
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