Theorem grpss 18512

Description: Show that a structure extending a constructed group (e.g., a ring) is also a group. This allows us to prove that a constructed potential ring 𝑅 is a group before we know that it is also a ring. (Theorem ringgrp 19703, 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	⊢ 𝐺 ⊆ 𝑅
grpss.f	⊢ Fun 𝑅

Assertion

Ref	Expression
grpss	⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)

Proof of Theorem grpss

Step	Hyp	Ref	Expression
1		grpss.r	. . . 4 ⊢ 𝑅 ∈ V
2		grpss.f	. . . 4 ⊢ Fun 𝑅
3		grpss.s	. . . 4 ⊢ 𝐺 ⊆ 𝑅
4		baseid 16843	. . . 4 ⊢ Base = Slot (Base‘ndx)
5		opex 5373	. . . . . 6 ⊢ ⟨(Base‘ndx), 𝐵⟩ ∈ V
6	5	prid1 4695	. . . . 5 ⊢ ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
7		grpss.g	. . . . 5 ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
8	6, 7	eleqtrri 2838	. . . 4 ⊢ ⟨(Base‘ndx), 𝐵⟩ ∈ 𝐺
9	1, 2, 3, 4, 8	strss 16836	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
10		plusgid 16915	. . . 4 ⊢ +g = Slot (+g‘ndx)
11		opex 5373	. . . . . 6 ⊢ ⟨(+g‘ndx), + ⟩ ∈ V
12	11	prid2 4696	. . . . 5 ⊢ ⟨(+g‘ndx), + ⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}
13	12, 7	eleqtrri 2838	. . . 4 ⊢ ⟨(+g‘ndx), + ⟩ ∈ 𝐺
14	1, 2, 3, 10, 13	strss 16836	. . 3 ⊢ (+g‘𝑅) = (+g‘𝐺)
15	9, 14	grpprop 18510	. 2 ⊢ (𝑅 ∈ Grp ↔ 𝐺 ∈ Grp)
16	15	bicomi 223	1 ⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  {cpr 4560  ⟨cop 4564  Fun wfun 6412  ‘cfv 6418  ndxcnx 16822  Basecbs 16840  +_gcplusg 16888  Grpcgrp 18492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-1cn 10860  ax-addcl 10862

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-nn 11904  df-2 11966  df-slot 16811  df-ndx 16823  df-base 16841  df-plusg 16901  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495

This theorem is referenced by: (None)