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Theorem lsmsnorb 30999
 Description: The sumset of a group with a single element is the element's orbit by the group action. See gaorb 18429. (Contributed by Thierry Arnoux, 21-Jan-2024.)
Hypotheses
Ref Expression
lsmsnorb.1 𝐵 = (Base‘𝐺)
lsmsnorb.2 + = (+g𝐺)
lsmsnorb.3 = (LSSum‘𝐺)
lsmsnorb.4 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
lsmsnorb.5 (𝜑𝐺 ∈ Mnd)
lsmsnorb.6 (𝜑𝐴𝐵)
lsmsnorb.7 (𝜑𝑋𝐵)
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmsnorb.5 . . . 4 (𝜑𝐺 ∈ Mnd)
2 lsmsnorb.6 . . . 4 (𝜑𝐴𝐵)
3 lsmsnorb.7 . . . . 5 (𝜑𝑋𝐵)
43snssd 4702 . . . 4 (𝜑 → {𝑋} ⊆ 𝐵)
5 lsmsnorb.1 . . . . 5 𝐵 = (Base‘𝐺)
6 lsmsnorb.3 . . . . 5 = (LSSum‘𝐺)
75, 6lsmssv 18760 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐴𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝐴 {𝑋}) ⊆ 𝐵)
81, 2, 4, 7syl3anc 1368 . . 3 (𝜑 → (𝐴 {𝑋}) ⊆ 𝐵)
98sselda 3915 . 2 ((𝜑𝑘 ∈ (𝐴 {𝑋})) → 𝑘𝐵)
10 df-ec 8274 . . . 4 [𝑋] = ( “ {𝑋})
11 imassrn 5907 . . . . . 6 ( “ {𝑋}) ⊆ ran
12 lsmsnorb.4 . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
1312rneqi 5771 . . . . . . 7 ran = ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
14 rnopab 5790 . . . . . . . 8 ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} = {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
15 vex 3444 . . . . . . . . . . . . . 14 𝑥 ∈ V
16 vex 3444 . . . . . . . . . . . . . 14 𝑦 ∈ V
1715, 16prss 4713 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
1817biimpri 231 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐵 → (𝑥𝐵𝑦𝐵))
1918simprd 499 . . . . . . . . . . 11 ({𝑥, 𝑦} ⊆ 𝐵𝑦𝐵)
2019adantr 484 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦𝐵)
2120exlimiv 1931 . . . . . . . . 9 (∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦𝐵)
2221abssi 3997 . . . . . . . 8 {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
2314, 22eqsstri 3949 . . . . . . 7 ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
2413, 23eqsstri 3949 . . . . . 6 ran 𝐵
2511, 24sstri 3924 . . . . 5 ( “ {𝑋}) ⊆ 𝐵
2625a1i 11 . . . 4 (𝜑 → ( “ {𝑋}) ⊆ 𝐵)
2710, 26eqsstrid 3963 . . 3 (𝜑 → [𝑋] 𝐵)
2827sselda 3915 . 2 ((𝜑𝑘 ∈ [𝑋] ) → 𝑘𝐵)
293anim1i 617 . . . . . 6 ((𝜑𝑘𝐵) → (𝑋𝐵𝑘𝐵))
3029biantrurd 536 . . . . 5 ((𝜑𝑘𝐵) → (∃𝐴 ( + 𝑋) = 𝑘 ↔ ((𝑋𝐵𝑘𝐵) ∧ ∃𝐴 ( + 𝑋) = 𝑘)))
31 df-3an 1086 . . . . 5 ((𝑋𝐵𝑘𝐵 ∧ ∃𝐴 ( + 𝑋) = 𝑘) ↔ ((𝑋𝐵𝑘𝐵) ∧ ∃𝐴 ( + 𝑋) = 𝑘))
3230, 31syl6bbr 292 . . . 4 ((𝜑𝑘𝐵) → (∃𝐴 ( + 𝑋) = 𝑘 ↔ (𝑋𝐵𝑘𝐵 ∧ ∃𝐴 ( + 𝑋) = 𝑘)))
3312gaorb 18429 . . . 4 (𝑋 𝑘 ↔ (𝑋𝐵𝑘𝐵 ∧ ∃𝐴 ( + 𝑋) = 𝑘))
3432, 33syl6rbbr 293 . . 3 ((𝜑𝑘𝐵) → (𝑋 𝑘 ↔ ∃𝐴 ( + 𝑋) = 𝑘))
35 vex 3444 . . . 4 𝑘 ∈ V
363adantr 484 . . . 4 ((𝜑𝑘𝐵) → 𝑋𝐵)
37 elecg 8315 . . . 4 ((𝑘 ∈ V ∧ 𝑋𝐵) → (𝑘 ∈ [𝑋] 𝑋 𝑘))
3835, 36, 37sylancr 590 . . 3 ((𝜑𝑘𝐵) → (𝑘 ∈ [𝑋] 𝑋 𝑘))
39 lsmsnorb.2 . . . . 5 + = (+g𝐺)
401adantr 484 . . . . 5 ((𝜑𝑘𝐵) → 𝐺 ∈ Mnd)
412adantr 484 . . . . 5 ((𝜑𝑘𝐵) → 𝐴𝐵)
425, 39, 6, 40, 41, 36elgrplsmsn 30998 . . . 4 ((𝜑𝑘𝐵) → (𝑘 ∈ (𝐴 {𝑋}) ↔ ∃𝐴 𝑘 = ( + 𝑋)))
43 eqcom 2805 . . . . 5 (𝑘 = ( + 𝑋) ↔ ( + 𝑋) = 𝑘)
4443rexbii 3210 . . . 4 (∃𝐴 𝑘 = ( + 𝑋) ↔ ∃𝐴 ( + 𝑋) = 𝑘)
4542, 44syl6bb 290 . . 3 ((𝜑𝑘𝐵) → (𝑘 ∈ (𝐴 {𝑋}) ↔ ∃𝐴 ( + 𝑋) = 𝑘))
4634, 38, 453bitr4rd 315 . 2 ((𝜑𝑘𝐵) → (𝑘 ∈ (𝐴 {𝑋}) ↔ 𝑘 ∈ [𝑋] ))
479, 28, 46eqrdav 2797 1 (𝜑 → (𝐴 {𝑋}) = [𝑋] )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  {csn 4525  {cpr 4527   class class class wbr 5030  {copab 5092  ran crn 5520   “ cima 5522  ‘cfv 6324  (class class class)co 7135  [cec 8270  Basecbs 16475  +gcplusg 16557  Mndcmnd 17903  LSSumclsm 18751 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-ec 8274  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-lsm 18753 This theorem is referenced by: (None)
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