Theorem lsmsnorb 30967

Description: The sumset of a group with a single element is the element's orbit by the group action. See gaorb 18433. (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 4739	. . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
5		lsmsnorb.1	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
6		lsmsnorb.3	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
7	5, 6	lsmssv 18764	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝐴 ⊕ {𝑋}) ⊆ 𝐵)
8	1, 2, 4, 7	syl3anc 1366	. . 3 ⊢ (𝜑 → (𝐴 ⊕ {𝑋}) ⊆ 𝐵)
9	8	sselda 3964	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ⊕ {𝑋})) → 𝑘 ∈ 𝐵)
10		df-ec 8288	. . . 4 ⊢ [𝑋] ∼ = ( ∼ “ {𝑋})
11		imassrn 5937	. . . . . 6 ⊢ ( ∼ “ {𝑋}) ⊆ ran ∼
12		lsmsnorb.4	. . . . . . . 8 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
13	12	rneqi 5804	. . . . . . 7 ⊢ ran ∼ = ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
14		rnopab 5823	. . . . . . . 8 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} = {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
15		vex 3496	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
16		vex 3496	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
17	15, 16	prss 4750	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
18	17	biimpri 230	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
19	18	simprd 498	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → 𝑦 ∈ 𝐵)
20	19	adantr 483	. . . . . . . . . 10 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
21	20	exlimiv 1930	. . . . . . . . 9 ⊢ (∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
22	21	abssi 4043	. . . . . . . 8 ⊢ {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
23	14, 22	eqsstri 3998	. . . . . . 7 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
24	13, 23	eqsstri 3998	. . . . . 6 ⊢ ran ∼ ⊆ 𝐵
25	11, 24	sstri 3973	. . . . 5 ⊢ ( ∼ “ {𝑋}) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → ( ∼ “ {𝑋}) ⊆ 𝐵)
27	10, 26	eqsstrid 4012	. . 3 ⊢ (𝜑 → [𝑋] ∼ ⊆ 𝐵)
28	27	sselda 3964	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ [𝑋] ∼ ) → 𝑘 ∈ 𝐵)
29	3	anim1i 616	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵))
30	29	biantrurd 535	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵) ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)))
31		df-3an 1084	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘) ↔ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵) ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
32	30, 31	syl6bbr 291	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘 ↔ (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)))
33	12	gaorb 18433	. . . 4 ⊢ (𝑋 ∼ 𝑘 ↔ (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
34	32, 33	syl6rbbr 292	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋 ∼ 𝑘 ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
35		vex 3496	. . . 4 ⊢ 𝑘 ∈ V
36	3	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑋 ∈ 𝐵)
37		elecg 8329	. . . 4 ⊢ ((𝑘 ∈ V ∧ 𝑋 ∈ 𝐵) → (𝑘 ∈ [𝑋] ∼ ↔ 𝑋 ∼ 𝑘))
38	35, 36, 37	sylancr 589	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ [𝑋] ∼ ↔ 𝑋 ∼ 𝑘))
39		lsmsnorb.2	. . . . 5 ⊢ + = (+g‘𝐺)
40	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ Mnd)
41	2	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
42	5, 39, 6, 40, 41, 36	elgrplsmsn 30966	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃ℎ ∈ 𝐴 𝑘 = (ℎ + 𝑋)))
43		eqcom 2827	. . . . 5 ⊢ (𝑘 = (ℎ + 𝑋) ↔ (ℎ + 𝑋) = 𝑘)
44	43	rexbii 3246	. . . 4 ⊢ (∃ℎ ∈ 𝐴 𝑘 = (ℎ + 𝑋) ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)
45	42, 44	syl6bb 289	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
46	34, 38, 45	3bitr4rd 314	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ 𝑘 ∈ [𝑋] ∼ ))
47	9, 28, 46	eqrdav 2819	1 ⊢ (𝜑 → (𝐴 ⊕ {𝑋}) = [𝑋] ∼ )

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2798  ∃wrex 3138  Vcvv 3493   ⊆ wss 3933  {csn 4564  {cpr 4566   class class class wbr 5063  {copab 5125  ran crn 5553   “ cima 5555  ‘cfv 6352  (class class class)co 7153  [cec 8284  Basecbs 16479  +_gcplusg 16561  Mndcmnd 17907  LSSumclsm 18755

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327  ax-un 7458

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3495  df-sbc 3771  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-iun 4918  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5457  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7156  df-oprab 7157  df-mpo 7158  df-1st 7686  df-2nd 7687  df-ec 8288  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-lsm 18757

This theorem is referenced by: (None)