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Mirrors > Home > MPE Home > Th. List > grpss | Structured version Visualization version GIF version |
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 19567, on the other hand, requires that we know in advance that 𝑅 is a ring.) (Contributed by NM, 11-Oct-2013.) |
Ref | Expression |
---|---|
grpss.g | ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
grpss.r | ⊢ 𝑅 ∈ V |
grpss.s | ⊢ 𝐺 ⊆ 𝑅 |
grpss.f | ⊢ Fun 𝑅 |
Ref | Expression |
---|---|
grpss | ⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpss.r | . . . 4 ⊢ 𝑅 ∈ V | |
2 | grpss.f | . . . 4 ⊢ Fun 𝑅 | |
3 | grpss.s | . . . 4 ⊢ 𝐺 ⊆ 𝑅 | |
4 | baseid 16763 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
5 | opex 5348 | . . . . . 6 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ V | |
6 | 5 | prid1 4678 | . . . . 5 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
7 | grpss.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} | |
8 | 6, 7 | eleqtrri 2837 | . . . 4 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ 𝐺 |
9 | 1, 2, 3, 4, 8 | strss 16757 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝐺) |
10 | plusgid 16829 | . . . 4 ⊢ +g = Slot (+g‘ndx) | |
11 | opex 5348 | . . . . . 6 ⊢ 〈(+g‘ndx), + 〉 ∈ V | |
12 | 11 | prid2 4679 | . . . . 5 ⊢ 〈(+g‘ndx), + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
13 | 12, 7 | eleqtrri 2837 | . . . 4 ⊢ 〈(+g‘ndx), + 〉 ∈ 𝐺 |
14 | 1, 2, 3, 10, 13 | strss 16757 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝐺) |
15 | 9, 14 | grpprop 18383 | . 2 ⊢ (𝑅 ∈ Grp ↔ 𝐺 ∈ Grp) |
16 | 15 | bicomi 227 | 1 ⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 {cpr 4543 〈cop 4547 Fun wfun 6374 ‘cfv 6380 ndxcnx 16744 Basecbs 16760 +gcplusg 16802 Grpcgrp 18365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-1cn 10787 ax-addcl 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-nn 11831 df-2 11893 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 |
This theorem is referenced by: (None) |
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