Theorem elgrplsmsn 33878

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 4746	. . 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 19815	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
9	1, 2, 4, 8	syl3anc 1398	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10		oveq2 7416	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
11	10	eqeq2d 2771	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
12	11	rexsng 4636	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
13	3, 12	syl 18	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
14	13	rexbidv 3186	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3898  {csn 4583  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  +_gcplusg 17389  LSSumclsm 19809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-lsm 19811

This theorem is used by:  lsmsnorb  33879  lsmsnpridl  33884  mxidlprm  33928