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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 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.) |
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 17091 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
5 | opex 5422 | . . . . . 6 ⊢ ⟨(Base‘ndx), 𝐵⟩ ∈ V | |
6 | 5 | prid1 4724 | . . . . 5 ⊢ ⟨(Base‘ndx), 𝐵⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩} |
7 | grpss.g | . . . . 5 ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩} | |
8 | 6, 7 | eleqtrri 2833 | . . . 4 ⊢ ⟨(Base‘ndx), 𝐵⟩ ∈ 𝐺 |
9 | 1, 2, 3, 4, 8 | strss 17084 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝐺) |
10 | plusgid 17165 | . . . 4 ⊢ +g = Slot (+g‘ndx) | |
11 | opex 5422 | . . . . . 6 ⊢ ⟨(+g‘ndx), + ⟩ ∈ V | |
12 | 11 | prid2 4725 | . . . . 5 ⊢ ⟨(+g‘ndx), + ⟩ ∈ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩} |
13 | 12, 7 | eleqtrri 2833 | . . . 4 ⊢ ⟨(+g‘ndx), + ⟩ ∈ 𝐺 |
14 | 1, 2, 3, 10, 13 | strss 17084 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝐺) |
15 | 9, 14 | grpprop 18771 | . 2 ⊢ (𝑅 ∈ Grp ↔ 𝐺 ∈ Grp) |
16 | 15 | bicomi 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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