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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 2728 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | oppgbas 19310 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ Grp → (Base‘𝑅) = (Base‘𝑂)) |
5 | eqidd 2729 | . 2 ⊢ (𝑅 ∈ Grp → (+g‘𝑂) = (+g‘𝑂)) | |
6 | eqid 2728 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 6 | oppgid 19317 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
8 | 7 | a1i 11 | . 2 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) = (0g‘𝑂)) |
9 | grpmnd 18904 | . . 3 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd) | |
10 | 1 | oppgmnd 19315 | . . 3 ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd) |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Mnd) |
12 | eqid 2728 | . . 3 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
13 | 2, 12 | grpinvcl 18951 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘𝑥) ∈ (Base‘𝑅)) |
14 | eqid 2728 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
15 | eqid 2728 | . . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
16 | 14, 1, 15 | oppgplus 19307 | . . 3 ⊢ (((invg‘𝑅)‘𝑥)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑥)) |
17 | 2, 14, 6, 12 | grprinv 18954 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑥)) = (0g‘𝑅)) |
18 | 16, 17 | eqtrid 2780 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (((invg‘𝑅)‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
19 | 4, 5, 8, 11, 13, 18 | isgrpd2 18920 | 1 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 Mndcmnd 18701 Grpcgrp 18897 invgcminusg 18898 oppgcoppg 19303 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-oppg 19304 |
This theorem is referenced by: oppggrpb 19319 oppginv 19320 invoppggim 19321 oppgtgp 24022 |
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