Theorem elgrplsmsn 33578

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 4747	. . 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 19682	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
9	1, 2, 4, 8	syl3anc 1392	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10		oveq2 7406	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
11	10	eqeq2d 2775	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
12	11	rexsng 4637	. . . 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 3188	. 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 1562   ∈ wcel 2144  ∃wrex 3088   ⊆ wss 3906  {csn 4584  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  LSSumclsm 19676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-lsm 19678

This theorem is referenced by:  lsmsnorb  33579  lsmsnpridl  33586  mxidlprm  33660