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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 18552, 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 15919 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
5 | opex 4932 | . . . . . 6 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ V | |
6 | 5 | prid1 4297 | . . . . 5 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
7 | grpss.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} | |
8 | 6, 7 | eleqtrri 2700 | . . . 4 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ 𝐺 |
9 | 1, 2, 3, 4, 8 | strss 15910 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝐺) |
10 | plusgid 15977 | . . . 4 ⊢ +g = Slot (+g‘ndx) | |
11 | opex 4932 | . . . . . 6 ⊢ 〈(+g‘ndx), + 〉 ∈ V | |
12 | 11 | prid2 4298 | . . . . 5 ⊢ 〈(+g‘ndx), + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
13 | 12, 7 | eleqtrri 2700 | . . . 4 ⊢ 〈(+g‘ndx), + 〉 ∈ 𝐺 |
14 | 1, 2, 3, 10, 13 | strss 15910 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝐺) |
15 | 9, 14 | grpprop 17438 | . 2 ⊢ (𝑅 ∈ Grp ↔ 𝐺 ∈ Grp) |
16 | 15 | bicomi 214 | 1 ⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {cpr 4179 〈cop 4183 Fun wfun 5882 ‘cfv 5888 ndxcnx 15854 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 |
This theorem is referenced by: (None) |
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