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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 2739 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | oppgbas 18937 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Grp → (Base‘𝑅) = (Base‘𝑂)) |
5 | eqidd 2740 | . 2 ⊢ (𝑅 ∈ Grp → (+g‘𝑂) = (+g‘𝑂)) | |
6 | eqid 2739 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 6 | oppgid 18944 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
8 | 7 | a1i 11 | . 2 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) = (0g‘𝑂)) |
9 | grpmnd 18565 | . . 3 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd) | |
10 | 1 | oppgmnd 18942 | . . 3 ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Mnd) |
12 | eqid 2739 | . . 3 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
13 | 2, 12 | grpinvcl 18608 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘𝑥) ∈ (Base‘𝑅)) |
14 | eqid 2739 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
15 | eqid 2739 | . . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
16 | 14, 1, 15 | oppgplus 18934 | . . 3 ⊢ (((invg‘𝑅)‘𝑥)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑥)) |
17 | 2, 14, 6, 12 | grprinv 18610 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑥)) = (0g‘𝑅)) |
18 | 16, 17 | eqtrid 2791 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (((invg‘𝑅)‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
19 | 4, 5, 8, 11, 13, 18 | isgrpd2 18580 | 1 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 +gcplusg 16943 0gc0g 17131 Mndcmnd 18366 Grpcgrp 18558 invgcminusg 18559 oppgcoppg 18930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-oppg 18931 |
This theorem is referenced by: oppggrpb 18946 oppginv 18947 invoppggim 18948 oppgtgp 23230 |
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