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Theorem grpss 18773
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 19974, 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 17091 . . . 4 Base = Slot (Base‘ndx)
5 opex 5422 . . . . . 6 ⟨(Base‘ndx), 𝐵⟩ ∈ V
65prid1 4724 . . . . 5 ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
7 grpss.g . . . . 5 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
86, 7eleqtrri 2833 . . . 4 ⟨(Base‘ndx), 𝐵⟩ ∈ 𝐺
91, 2, 3, 4, 8strss 17084 . . 3 (Base‘𝑅) = (Base‘𝐺)
10 plusgid 17165 . . . 4 +g = Slot (+g‘ndx)
11 opex 5422 . . . . . 6 ⟨(+g‘ndx), + ⟩ ∈ V
1211prid2 4725 . . . . 5 ⟨(+g‘ndx), + ⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
1312, 7eleqtrri 2833 . . . 4 ⟨(+g‘ndx), + ⟩ ∈ 𝐺
141, 2, 3, 10, 13strss 17084 . . 3 (+g𝑅) = (+g𝐺)
159, 14grpprop 18771 . 2 (𝑅 ∈ Grp ↔ 𝐺 ∈ Grp)
1615bicomi 223 1 (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  Vcvv 3444  wss 3911  {cpr 4589  cop 4593  Fun wfun 6491  cfv 6497  ndxcnx 17070  Basecbs 17088  +gcplusg 17138  Grpcgrp 18753
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-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-1cn 11114  ax-addcl 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-nn 12159  df-2 12221  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756
This theorem is referenced by: (None)
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