Theorem elgrplsmsn 33477

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 4721	. . 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 19610	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
9	1, 2, 4, 8	syl3anc 1380	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10		oveq2 7368	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
11	10	eqeq2d 2752	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
12	11	rexsng 4611	. . . 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 3165	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋)))
15	9, 14	bitrd 281	1 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121  ∃wrex 3065   ⊆ wss 3885  {csn 4558  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  LSSumclsm 19604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-lsm 19606

This theorem is referenced by:  lsmsnorb  33478  lsmsnpridl  33485  mxidlprm  33557