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	⊢ + = (+g‘𝐺)
elgrplsmsn.3	⊢ ⊕ = (LSSum‘𝐺)
elgrplsmsn.4	⊢ (𝜑 → 𝐺 ∈ 𝑉)
elgrplsmsn.5	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
elgrplsmsn.6	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
elgrplsmsn	⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		elgrplsmsn.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
2		elgrplsmsn.5	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		elgrplsmsn.6	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4	3	snssd 4692	. . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
5		elgrplsmsn.1	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
6		elgrplsmsn.2	. . . 4 ⊢ + = (+g‘𝐺)
7		elgrplsmsn.3	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
8	5, 6, 7	lsmelvalx 18817	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
9	1, 2, 4, 8	syl3anc 1369	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10		oveq2 7151	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
11	10	eqeq2d 2770	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
12	11	rexsng 4564	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
13	3, 12	syl 17	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
14	13	rexbidv 3219	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋)))
15	9, 14	bitrd 282	1 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋)))

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