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Theorem lsmsnorb 31575
Description: The sumset of a group with a single element is the element's orbit by the group action. See gaorb 18911. (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 4748 . . . 4 (𝜑 → {𝑋} ⊆ 𝐵)
5 lsmsnorb.1 . . . . 5 𝐵 = (Base‘𝐺)
6 lsmsnorb.3 . . . . 5 = (LSSum‘𝐺)
75, 6lsmssv 19246 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐴𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝐴 {𝑋}) ⊆ 𝐵)
81, 2, 4, 7syl3anc 1370 . . 3 (𝜑 → (𝐴 {𝑋}) ⊆ 𝐵)
98sselda 3926 . 2 ((𝜑𝑘 ∈ (𝐴 {𝑋})) → 𝑘𝐵)
10 df-ec 8483 . . . 4 [𝑋] = ( “ {𝑋})
11 imassrn 5979 . . . . . 6 ( “ {𝑋}) ⊆ ran
12 lsmsnorb.4 . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
1312rneqi 5845 . . . . . . 7 ran = ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
14 rnopab 5862 . . . . . . . 8 ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} = {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
15 vex 3435 . . . . . . . . . . . . . 14 𝑥 ∈ V
16 vex 3435 . . . . . . . . . . . . . 14 𝑦 ∈ V
1715, 16prss 4759 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
1817biimpri 227 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐵 → (𝑥𝐵𝑦𝐵))
1918simprd 496 . . . . . . . . . . 11 ({𝑥, 𝑦} ⊆ 𝐵𝑦𝐵)
2019adantr 481 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦𝐵)
2120exlimiv 1937 . . . . . . . . 9 (∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦𝐵)
2221abssi 4008 . . . . . . . 8 {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
2314, 22eqsstri 3960 . . . . . . 7 ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
2413, 23eqsstri 3960 . . . . . 6 ran 𝐵
2511, 24sstri 3935 . . . . 5 ( “ {𝑋}) ⊆ 𝐵
2625a1i 11 . . . 4 (𝜑 → ( “ {𝑋}) ⊆ 𝐵)
2710, 26eqsstrid 3974 . . 3 (𝜑 → [𝑋] 𝐵)
2827sselda 3926 . 2 ((𝜑𝑘 ∈ [𝑋] ) → 𝑘𝐵)
2912gaorb 18911 . . . 4 (𝑋 𝑘 ↔ (𝑋𝐵𝑘𝐵 ∧ ∃𝐴 ( + 𝑋) = 𝑘))
303anim1i 615 . . . . . 6 ((𝜑𝑘𝐵) → (𝑋𝐵𝑘𝐵))
3130biantrurd 533 . . . . 5 ((𝜑𝑘𝐵) → (∃𝐴 ( + 𝑋) = 𝑘 ↔ ((𝑋𝐵𝑘𝐵) ∧ ∃𝐴 ( + 𝑋) = 𝑘)))
32 df-3an 1088 . . . . 5 ((𝑋𝐵𝑘𝐵 ∧ ∃𝐴 ( + 𝑋) = 𝑘) ↔ ((𝑋𝐵𝑘𝐵) ∧ ∃𝐴 ( + 𝑋) = 𝑘))
3331, 32bitr4di 289 . . . 4 ((𝜑𝑘𝐵) → (∃𝐴 ( + 𝑋) = 𝑘 ↔ (𝑋𝐵𝑘𝐵 ∧ ∃𝐴 ( + 𝑋) = 𝑘)))
3429, 33bitr4id 290 . . 3 ((𝜑𝑘𝐵) → (𝑋 𝑘 ↔ ∃𝐴 ( + 𝑋) = 𝑘))
35 vex 3435 . . . 4 𝑘 ∈ V
363adantr 481 . . . 4 ((𝜑𝑘𝐵) → 𝑋𝐵)
37 elecg 8524 . . . 4 ((𝑘 ∈ V ∧ 𝑋𝐵) → (𝑘 ∈ [𝑋] 𝑋 𝑘))
3835, 36, 37sylancr 587 . . 3 ((𝜑𝑘𝐵) → (𝑘 ∈ [𝑋] 𝑋 𝑘))
39 lsmsnorb.2 . . . . 5 + = (+g𝐺)
401adantr 481 . . . . 5 ((𝜑𝑘𝐵) → 𝐺 ∈ Mnd)
412adantr 481 . . . . 5 ((𝜑𝑘𝐵) → 𝐴𝐵)
425, 39, 6, 40, 41, 36elgrplsmsn 31574 . . . 4 ((𝜑𝑘𝐵) → (𝑘 ∈ (𝐴 {𝑋}) ↔ ∃𝐴 𝑘 = ( + 𝑋)))
43 eqcom 2747 . . . . 5 (𝑘 = ( + 𝑋) ↔ ( + 𝑋) = 𝑘)
4443rexbii 3180 . . . 4 (∃𝐴 𝑘 = ( + 𝑋) ↔ ∃𝐴 ( + 𝑋) = 𝑘)
4542, 44bitrdi 287 . . 3 ((𝜑𝑘𝐵) → (𝑘 ∈ (𝐴 {𝑋}) ↔ ∃𝐴 ( + 𝑋) = 𝑘))
4634, 38, 453bitr4rd 312 . 2 ((𝜑𝑘𝐵) → (𝑘 ∈ (𝐴 {𝑋}) ↔ 𝑘 ∈ [𝑋] ))
479, 28, 46eqrdav 2739 1 (𝜑 → (𝐴 {𝑋}) = [𝑋] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1542  wex 1786  wcel 2110  {cab 2717  wrex 3067  Vcvv 3431  wss 3892  {csn 4567  {cpr 4569   class class class wbr 5079  {copab 5141  ran crn 5591  cima 5593  cfv 6432  (class class class)co 7271  [cec 8479  Basecbs 16910  +gcplusg 16960  Mndcmnd 18383  LSSumclsm 19237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-ec 8483  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-lsm 19239
This theorem is referenced by:  lsmsnorb2  31576
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