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Theorem elgrplsmsn 32219
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 4770 . . 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 19427 . . 3 ((𝐺𝑉𝐴𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
91, 2, 4, 8syl3anc 1372 . 2 (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10 oveq2 7366 . . . . . 6 (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
1110eqeq2d 2744 . . . . 5 (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
1211rexsng 4636 . . . 4 (𝑋𝐵 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
133, 12syl 17 . . 3 (𝜑 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
1413rexbidv 3172 . 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 205   = wceq 1542  wcel 2107  wrex 3070  wss 3911  {csn 4587  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  LSSumclsm 19421
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-lsm 19423
This theorem is referenced by:  lsmsnorb  32220  lsmsnpridl  32227  mxidlprm  32285
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