![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > oppggrp | Structured version Visualization version GIF version |
Description: The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
oppgbas.1 | ⊢ 𝑂 = (oppg‘𝑅) |
Ref | Expression |
---|---|
oppggrp | ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppgbas.1 | . . . 4 ⊢ 𝑂 = (oppg‘𝑅) | |
2 | eqid 2725 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | oppgbas 19320 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Grp → (Base‘𝑅) = (Base‘𝑂)) |
5 | eqidd 2726 | . 2 ⊢ (𝑅 ∈ Grp → (+g‘𝑂) = (+g‘𝑂)) | |
6 | eqid 2725 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 6 | oppgid 19327 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
8 | 7 | a1i 11 | . 2 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) = (0g‘𝑂)) |
9 | grpmnd 18910 | . . 3 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd) | |
10 | 1 | oppgmnd 19325 | . . 3 ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Mnd) |
12 | eqid 2725 | . . 3 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
13 | 2, 12 | grpinvcl 18957 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘𝑥) ∈ (Base‘𝑅)) |
14 | eqid 2725 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
15 | eqid 2725 | . . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
16 | 14, 1, 15 | oppgplus 19317 | . . 3 ⊢ (((invg‘𝑅)‘𝑥)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑥)) |
17 | 2, 14, 6, 12 | grprinv 18960 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑥)) = (0g‘𝑅)) |
18 | 16, 17 | eqtrid 2777 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (((invg‘𝑅)‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
19 | 4, 5, 8, 11, 13, 18 | isgrpd2 18926 | 1 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17188 +gcplusg 17241 0gc0g 17429 Mndcmnd 18702 Grpcgrp 18903 invgcminusg 18904 oppgcoppg 19313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-plusg 17254 df-0g 17431 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18906 df-minusg 18907 df-oppg 19314 |
This theorem is referenced by: oppggrpb 19329 oppginv 19330 invoppggim 19331 oppgtgp 24051 |
Copyright terms: Public domain | W3C validator |