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Theorem grpss 18910
Description: Show that a structure extending a constructed group (e.g., a ring) is also a group. This allows to prove that a constructed potential ring 𝑅 is a group before we know that it is also a ring. (Theorem ringgrp 20177, on the other hand, requires that we know in advance that 𝑅 is a ring.) (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
grpss.g 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
grpss.r 𝑅 ∈ V
grpss.s 𝐺𝑅
grpss.f Fun 𝑅
Assertion
Ref Expression
grpss (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)

Proof of Theorem grpss
StepHypRef Expression
1 grpss.r . . . 4 𝑅 ∈ V
2 grpss.f . . . 4 Fun 𝑅
3 grpss.s . . . 4 𝐺𝑅
4 baseid 17182 . . . 4 Base = Slot (Base‘ndx)
5 opex 5466 . . . . . 6 ⟨(Base‘ndx), 𝐵⟩ ∈ V
65prid1 4767 . . . . 5 ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
7 grpss.g . . . . 5 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
86, 7eleqtrri 2828 . . . 4 ⟨(Base‘ndx), 𝐵⟩ ∈ 𝐺
91, 2, 3, 4, 8strss 17175 . . 3 (Base‘𝑅) = (Base‘𝐺)
10 plusgid 17259 . . . 4 +g = Slot (+g‘ndx)
11 opex 5466 . . . . . 6 ⟨(+g‘ndx), + ⟩ ∈ V
1211prid2 4768 . . . . 5 ⟨(+g‘ndx), + ⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
1312, 7eleqtrri 2828 . . . 4 ⟨(+g‘ndx), + ⟩ ∈ 𝐺
141, 2, 3, 10, 13strss 17175 . . 3 (+g𝑅) = (+g𝐺)
159, 14grpprop 18908 . 2 (𝑅 ∈ Grp ↔ 𝐺 ∈ Grp)
1615bicomi 223 1 (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wcel 2099  Vcvv 3471  wss 3947  {cpr 4631  cop 4635  Fun wfun 6542  cfv 6548  ndxcnx 17161  Basecbs 17179  +gcplusg 17232  Grpcgrp 18889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-1cn 11196  ax-addcl 11198
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-nn 12243  df-2 12305  df-slot 17150  df-ndx 17162  df-base 17180  df-plusg 17245  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18892
This theorem is referenced by: (None)
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