Theorem lsmsnorb 32775

Description: The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19212. (Contributed by Thierry Arnoux, 21-Jan-2024.)

Hypotheses

Ref	Expression
lsmsnorb.1	⊢ 𝐵 = (Base‘𝐺)
lsmsnorb.2	⊢ + = (+g‘𝐺)
lsmsnorb.3	⊢ ⊕ = (LSSum‘𝐺)
lsmsnorb.4	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
lsmsnorb.5	⊢ (𝜑 → 𝐺 ∈ Mnd)
lsmsnorb.6	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
lsmsnorb.7	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
lsmsnorb	⊢ (𝜑 → (𝐴 ⊕ {𝑋}) = [𝑋] ∼ )

Distinct variable groups:   + ,𝑔,𝑥,𝑦   𝐴,𝑔,𝑥,𝑦   𝑥,𝐵,𝑦   𝑔,𝑋,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑔)   𝐵(𝑔)   ⊕ (𝑥,𝑦,𝑔)   ∼ (𝑥,𝑦,𝑔)   𝐺(𝑥,𝑦,𝑔)

Proof of Theorem lsmsnorb

Dummy variables 𝑘 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsmsnorb.5	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		lsmsnorb.6	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		lsmsnorb.7	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4	3	snssd 4811	. . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
5		lsmsnorb.1	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
6		lsmsnorb.3	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
7	5, 6	lsmssv 19552	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝐴 ⊕ {𝑋}) ⊆ 𝐵)
8	1, 2, 4, 7	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝐴 ⊕ {𝑋}) ⊆ 𝐵)
9	8	sselda 3981	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ⊕ {𝑋})) → 𝑘 ∈ 𝐵)
10		df-ec 8707	. . . 4 ⊢ [𝑋] ∼ = ( ∼ “ {𝑋})
11		imassrn 6069	. . . . . 6 ⊢ ( ∼ “ {𝑋}) ⊆ ran ∼
12		lsmsnorb.4	. . . . . . . 8 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
13	12	rneqi 5935	. . . . . . 7 ⊢ ran ∼ = ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
14		rnopab 5952	. . . . . . . 8 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} = {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
15		vex 3476	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
16		vex 3476	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
17	15, 16	prss 4822	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
18	17	biimpri 227	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
19	18	simprd 494	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → 𝑦 ∈ 𝐵)
20	19	adantr 479	. . . . . . . . . 10 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
21	20	exlimiv 1931	. . . . . . . . 9 ⊢ (∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
22	21	abssi 4066	. . . . . . . 8 ⊢ {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
23	14, 22	eqsstri 4015	. . . . . . 7 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
24	13, 23	eqsstri 4015	. . . . . 6 ⊢ ran ∼ ⊆ 𝐵
25	11, 24	sstri 3990	. . . . 5 ⊢ ( ∼ “ {𝑋}) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → ( ∼ “ {𝑋}) ⊆ 𝐵)
27	10, 26	eqsstrid 4029	. . 3 ⊢ (𝜑 → [𝑋] ∼ ⊆ 𝐵)
28	27	sselda 3981	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ [𝑋] ∼ ) → 𝑘 ∈ 𝐵)
29	12	gaorb 19212	. . . 4 ⊢ (𝑋 ∼ 𝑘 ↔ (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
30	3	anim1i 613	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵))
31	30	biantrurd 531	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵) ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)))
32		df-3an 1087	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘) ↔ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵) ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
33	31, 32	bitr4di 288	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘 ↔ (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)))
34	29, 33	bitr4id 289	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋 ∼ 𝑘 ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
35		vex 3476	. . . 4 ⊢ 𝑘 ∈ V
36	3	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑋 ∈ 𝐵)
37		elecg 8748	. . . 4 ⊢ ((𝑘 ∈ V ∧ 𝑋 ∈ 𝐵) → (𝑘 ∈ [𝑋] ∼ ↔ 𝑋 ∼ 𝑘))
38	35, 36, 37	sylancr 585	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ [𝑋] ∼ ↔ 𝑋 ∼ 𝑘))
39		lsmsnorb.2	. . . . 5 ⊢ + = (+g‘𝐺)
40	1	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ Mnd)
41	2	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
42	5, 39, 6, 40, 41, 36	elgrplsmsn 32774	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃ℎ ∈ 𝐴 𝑘 = (ℎ + 𝑋)))
43		eqcom 2737	. . . . 5 ⊢ (𝑘 = (ℎ + 𝑋) ↔ (ℎ + 𝑋) = 𝑘)
44	43	rexbii 3092	. . . 4 ⊢ (∃ℎ ∈ 𝐴 𝑘 = (ℎ + 𝑋) ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)
45	42, 44	bitrdi 286	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
46	34, 38, 45	3bitr4rd 311	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ 𝑘 ∈ [𝑋] ∼ ))
47	9, 28, 46	eqrdav 2729	1 ⊢ (𝜑 → (𝐴 ⊕ {𝑋}) = [𝑋] ∼ )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {cab 2707  ∃wrex 3068  Vcvv 3472   ⊆ wss 3947  {csn 4627  {cpr 4629   class class class wbr 5147  {copab 5209  ran crn 5676   “ cima 5678  ‘cfv 6542  (class class class)co 7411  [cec 8703  Basecbs 17148  +_gcplusg 17201  Mndcmnd 18659  LSSumclsm 19543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-ec 8707  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-lsm 19545

This theorem is referenced by:  lsmsnorb2  32776