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Theorem lsmsnorb 32501
Description: The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19171. (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 4813 . . . 4 (𝜑 → {𝑋} ⊆ 𝐵)
5 lsmsnorb.1 . . . . 5 𝐵 = (Base‘𝐺)
6 lsmsnorb.3 . . . . 5 = (LSSum‘𝐺)
75, 6lsmssv 19511 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐴𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝐴 {𝑋}) ⊆ 𝐵)
81, 2, 4, 7syl3anc 1372 . . 3 (𝜑 → (𝐴 {𝑋}) ⊆ 𝐵)
98sselda 3983 . 2 ((𝜑𝑘 ∈ (𝐴 {𝑋})) → 𝑘𝐵)
10 df-ec 8705 . . . 4 [𝑋] = ( “ {𝑋})
11 imassrn 6071 . . . . . 6 ( “ {𝑋}) ⊆ ran
12 lsmsnorb.4 . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
1312rneqi 5937 . . . . . . 7 ran = ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
14 rnopab 5954 . . . . . . . 8 ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} = {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}
15 vex 3479 . . . . . . . . . . . . . 14 𝑥 ∈ V
16 vex 3479 . . . . . . . . . . . . . 14 𝑦 ∈ V
1715, 16prss 4824 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
1817biimpri 227 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐵 → (𝑥𝐵𝑦𝐵))
1918simprd 497 . . . . . . . . . . 11 ({𝑥, 𝑦} ⊆ 𝐵𝑦𝐵)
2019adantr 482 . . . . . . . . . 10 (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦𝐵)
2120exlimiv 1934 . . . . . . . . 9 (∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦) → 𝑦𝐵)
2221abssi 4068 . . . . . . . 8 {𝑦 ∣ ∃𝑥({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
2314, 22eqsstri 4017 . . . . . . 7 ran {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)} ⊆ 𝐵
2413, 23eqsstri 4017 . . . . . 6 ran 𝐵
2511, 24sstri 3992 . . . . 5 ( “ {𝑋}) ⊆ 𝐵
2625a1i 11 . . . 4 (𝜑 → ( “ {𝑋}) ⊆ 𝐵)
2710, 26eqsstrid 4031 . . 3 (𝜑 → [𝑋] 𝐵)
2827sselda 3983 . 2 ((𝜑𝑘 ∈ [𝑋] ) → 𝑘𝐵)
2912gaorb 19171 . . . 4 (𝑋 𝑘 ↔ (𝑋𝐵𝑘𝐵 ∧ ∃𝐴 ( + 𝑋) = 𝑘))
303anim1i 616 . . . . . 6 ((𝜑𝑘𝐵) → (𝑋𝐵𝑘𝐵))
3130biantrurd 534 . . . . 5 ((𝜑𝑘𝐵) → (∃𝐴 ( + 𝑋) = 𝑘 ↔ ((𝑋𝐵𝑘𝐵) ∧ ∃𝐴 ( + 𝑋) = 𝑘)))
32 df-3an 1090 . . . . 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 3479 . . . 4 𝑘 ∈ V
363adantr 482 . . . 4 ((𝜑𝑘𝐵) → 𝑋𝐵)
37 elecg 8746 . . . 4 ((𝑘 ∈ V ∧ 𝑋𝐵) → (𝑘 ∈ [𝑋] 𝑋 𝑘))
3835, 36, 37sylancr 588 . . 3 ((𝜑𝑘𝐵) → (𝑘 ∈ [𝑋] 𝑋 𝑘))
39 lsmsnorb.2 . . . . 5 + = (+g𝐺)
401adantr 482 . . . . 5 ((𝜑𝑘𝐵) → 𝐺 ∈ Mnd)
412adantr 482 . . . . 5 ((𝜑𝑘𝐵) → 𝐴𝐵)
425, 39, 6, 40, 41, 36elgrplsmsn 32500 . . . 4 ((𝜑𝑘𝐵) → (𝑘 ∈ (𝐴 {𝑋}) ↔ ∃𝐴 𝑘 = ( + 𝑋)))
43 eqcom 2740 . . . . 5 (𝑘 = ( + 𝑋) ↔ ( + 𝑋) = 𝑘)
4443rexbii 3095 . . . 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 2732 1 (𝜑 → (𝐴 {𝑋}) = [𝑋] )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3071  Vcvv 3475  wss 3949  {csn 4629  {cpr 4631   class class class wbr 5149  {copab 5211  ran crn 5678  cima 5680  cfv 6544  (class class class)co 7409  [cec 8701  Basecbs 17144  +gcplusg 17197  Mndcmnd 18625  LSSumclsm 19502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-ec 8705  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-lsm 19504
This theorem is referenced by:  lsmsnorb2  32502
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