Theorem lsmsnorb 33928

Description: The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19501. (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 4747	. . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
5		lsmsnorb.1	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
6		lsmsnorb.3	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
7	5, 6	lsmssv 19837	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝐴 ⊕ {𝑋}) ⊆ 𝐵)
8	1, 2, 4, 7	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝐴 ⊕ {𝑋}) ⊆ 𝐵)
9	8	sselda 3931	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ⊕ {𝑋})) → 𝑘 ∈ 𝐵)
10		df-ec 8703	. . . 4 ⊢ [𝑋] ∼ = ( ∼ “ {𝑋})
11		imassrn 6065	. . . . . 6 ⊢ ( ∼ “ {𝑋}) ⊆ ran ∼
12		lsmsnorb.4	. . . . . . . 8 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
13	12	rneqi 5919	. . . . . . 7 ⊢ ran ∼ = ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
14		rnopab 5936	. . . . . . . 8 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} = {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}
15		vex 3455	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
16		vex 3455	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
17	15, 16	prss 4781	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
18	17	biimpri 231	. . . . . . . . . . . 12 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
19	18	simprd 501	. . . . . . . . . . 11 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → 𝑦 ∈ 𝐵)
20	19	adantr 486	. . . . . . . . . 10 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
21	20	exlimiv 1963	. . . . . . . . 9 ⊢ (∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
22	21	abssi 4016	. . . . . . . 8 ⊢ {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
23	14, 22	eqsstri 3977	. . . . . . 7 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
24	13, 23	eqsstri 3977	. . . . . 6 ⊢ ran ∼ ⊆ 𝐵
25	11, 24	sstri 3940	. . . . 5 ⊢ ( ∼ “ {𝑋}) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → ( ∼ “ {𝑋}) ⊆ 𝐵)
27	10, 26	eqsstrid 3969	. . 3 ⊢ (𝜑 → [𝑋] ∼ ⊆ 𝐵)
28	27	sselda 3931	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ [𝑋] ∼ ) → 𝑘 ∈ 𝐵)
29	12	gaorb 19501	. . . 4 ⊢ (𝑋 ∼ 𝑘 ↔ (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
30	3	anim1i 627	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵))
31	30	biantrurd 542	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵) ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)))
32		df-3an 1105	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘) ↔ ((𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵) ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
33	31, 32	bitr4di 292	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘 ↔ (𝑋 ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ∧ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)))
34	29, 33	bitr4id 293	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋 ∼ 𝑘 ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
35		vex 3455	. . . 4 ⊢ 𝑘 ∈ V
36	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑋 ∈ 𝐵)
37		elecg 8746	. . . 4 ⊢ ((𝑘 ∈ V ∧ 𝑋 ∈ 𝐵) → (𝑘 ∈ [𝑋] ∼ ↔ 𝑋 ∼ 𝑘))
38	35, 36, 37	sylancr 599	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ [𝑋] ∼ ↔ 𝑋 ∼ 𝑘))
39		lsmsnorb.2	. . . . 5 ⊢ + = (+g‘𝐺)
40	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ Mnd)
41	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
42	5, 39, 6, 40, 41, 36	elgrplsmsn 33927	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃ℎ ∈ 𝐴 𝑘 = (ℎ + 𝑋)))
43		eqcom 2768	. . . . 5 ⊢ (𝑘 = (ℎ + 𝑋) ↔ (ℎ + 𝑋) = 𝑘)
44	43	rexbii 3110	. . . 4 ⊢ (∃ℎ ∈ 𝐴 𝑘 = (ℎ + 𝑋) ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘)
45	42, 44	bitrdi 290	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃ℎ ∈ 𝐴 (ℎ + 𝑋) = 𝑘))
46	34, 38, 45	3bitr4rd 315	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ (𝐴 ⊕ {𝑋}) ↔ 𝑘 ∈ [𝑋] ∼ ))
47	9, 28, 46	eqrdav 2760	1 ⊢ (𝜑 → (𝐴 ⊕ {𝑋}) = [𝑋] ∼ )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  {csn 4584  {cpr 4586   class class class wbr 5103  {copab 5167  ran crn 5652   “ cima 5654  ‘cfv 6531  (class class class)co 7412  [cec 8699  Basecbs 17367  +_gcplusg 17408  Mndcmnd 18903  LSSumclsm 19828

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-ec 8703  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-lsm 19830

This theorem is used by:  lsmsnorb2  33929