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Theorem elgrplsmsn 33418
Description: Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.)
Hypotheses
Ref Expression
elgrplsmsn.1 𝐵 = (Base‘𝐺)
elgrplsmsn.2 + = (+g𝐺)
elgrplsmsn.3 = (LSSum‘𝐺)
elgrplsmsn.4 (𝜑𝐺𝑉)
elgrplsmsn.5 (𝜑𝐴𝐵)
elgrplsmsn.6 (𝜑𝑋𝐵)
Assertion
Ref Expression
elgrplsmsn (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))
Distinct variable groups:   𝑥, +   𝑥,𝐴   𝑥,𝐵   𝑥,𝐺   𝑥,𝑋   𝑥,𝑍   𝜑,𝑥
Allowed substitution hints:   (𝑥)   𝑉(𝑥)

Proof of Theorem elgrplsmsn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elgrplsmsn.4 . . 3 (𝜑𝐺𝑉)
2 elgrplsmsn.5 . . 3 (𝜑𝐴𝐵)
3 elgrplsmsn.6 . . . 4 (𝜑𝑋𝐵)
43snssd 4809 . . 3 (𝜑 → {𝑋} ⊆ 𝐵)
5 elgrplsmsn.1 . . . 4 𝐵 = (Base‘𝐺)
6 elgrplsmsn.2 . . . 4 + = (+g𝐺)
7 elgrplsmsn.3 . . . 4 = (LSSum‘𝐺)
85, 6, 7lsmelvalx 19658 . . 3 ((𝐺𝑉𝐴𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
91, 2, 4, 8syl3anc 1373 . 2 (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10 oveq2 7439 . . . . . 6 (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
1110eqeq2d 2748 . . . . 5 (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
1211rexsng 4676 . . . 4 (𝑋𝐵 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
133, 12syl 17 . . 3 (𝜑 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
1413rexbidv 3179 . 2 (𝜑 → (∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))
159, 14bitrd 279 1 (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  wss 3951  {csn 4626  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  LSSumclsm 19652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-lsm 19654
This theorem is referenced by:  lsmsnorb  33419  lsmsnpridl  33426  mxidlprm  33498
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