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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 19301, 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 16542 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
5 | opex 5355 | . . . . . 6 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ V | |
6 | 5 | prid1 4697 | . . . . 5 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
7 | grpss.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} | |
8 | 6, 7 | eleqtrri 2912 | . . . 4 ⊢ 〈(Base‘ndx), 𝐵〉 ∈ 𝐺 |
9 | 1, 2, 3, 4, 8 | strss 16533 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝐺) |
10 | plusgid 16595 | . . . 4 ⊢ +g = Slot (+g‘ndx) | |
11 | opex 5355 | . . . . . 6 ⊢ 〈(+g‘ndx), + 〉 ∈ V | |
12 | 11 | prid2 4698 | . . . . 5 ⊢ 〈(+g‘ndx), + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉} |
13 | 12, 7 | eleqtrri 2912 | . . . 4 ⊢ 〈(+g‘ndx), + 〉 ∈ 𝐺 |
14 | 1, 2, 3, 10, 13 | strss 16533 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝐺) |
15 | 9, 14 | grpprop 18118 | . 2 ⊢ (𝑅 ∈ Grp ↔ 𝐺 ∈ Grp) |
16 | 15 | bicomi 226 | 1 ⊢ (𝐺 ∈ Grp ↔ 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 {cpr 4568 〈cop 4572 Fun wfun 6348 ‘cfv 6354 ndxcnx 16479 Basecbs 16482 +gcplusg 16564 Grpcgrp 18102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-1cn 10594 ax-addcl 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-nn 11638 df-2 11699 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 |
This theorem is referenced by: (None) |
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