Theorem elgrplsmsn 32465

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 4811	. . 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 19501	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
9	1, 2, 4, 8	syl3anc 1372	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦)))
10		oveq2 7412	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋))
11	10	eqeq2d 2744	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋)))
12	11	rexsng 4677	. . . 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 3179	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋)))
15	9, 14	bitrd 279	1 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  ∃wrex 3071   ⊆ wss 3947  {csn 4627  ‘cfv 6540  (class class class)co 7404  Basecbs 17140  +_gcplusg 17193  LSSumclsm 19495

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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-lsm 19497

This theorem is referenced by:  lsmsnorb  32466  lsmsnpridl  32473  mxidlprm  32544