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Theorem elgrplsmsn 31084
 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 4692 . . 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 18817 . . 3 ((𝐺𝑉𝐴𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
91, 2, 4, 8syl3anc 1369 . 2 (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10 oveq2 7151 . . . . . 6 (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
1110eqeq2d 2770 . . . . 5 (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
1211rexsng 4564 . . . 4 (𝑋𝐵 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
133, 12syl 17 . . 3 (𝜑 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
1413rexbidv 3219 . 2 (𝜑 → (∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))
159, 14bitrd 282 1 (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539   ∈ wcel 2112  ∃wrex 3069   ⊆ wss 3854  {csn 4515  ‘cfv 6328  (class class class)co 7143  Basecbs 16526  +gcplusg 16608  LSSumclsm 18811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148  df-1st 7686  df-2nd 7687  df-lsm 18813 This theorem is referenced by:  lsmsnorb  31085  lsmsnpridl  31092  mxidlprm  31146
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