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Mirrors > Home > MPE Home > Th. List > oppginv | Structured version Visualization version GIF version |
Description: Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
oppgbas.1 | ⊢ 𝑂 = (oppg‘𝑅) |
oppginv.2 | ⊢ 𝐼 = (invg‘𝑅) |
Ref | Expression |
---|---|
oppginv | ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | oppginv.2 | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
3 | 1, 2 | grpinvf 17819 | . . 3 ⊢ (𝑅 ∈ Grp → 𝐼:(Base‘𝑅)⟶(Base‘𝑅)) |
4 | eqid 2824 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
5 | oppgbas.1 | . . . . . 6 ⊢ 𝑂 = (oppg‘𝑅) | |
6 | eqid 2824 | . . . . . 6 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
7 | 4, 5, 6 | oppgplus 18128 | . . . . 5 ⊢ ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)(𝐼‘𝑥)) |
8 | eqid 2824 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | 1, 4, 8, 2 | grprinv 17822 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(𝐼‘𝑥)) = (0g‘𝑅)) |
10 | 7, 9 | syl5eq 2872 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
11 | 10 | ralrimiva 3174 | . . 3 ⊢ (𝑅 ∈ Grp → ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
12 | 5 | oppggrp 18136 | . . . 4 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
13 | 5, 1 | oppgbas 18130 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
14 | 5, 8 | oppgid 18135 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑂) |
15 | eqid 2824 | . . . . 5 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
16 | 13, 6, 14, 15 | isgrpinv 17825 | . . . 4 ⊢ (𝑂 ∈ Grp → ((𝐼:(Base‘𝑅)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) ↔ (invg‘𝑂) = 𝐼)) |
17 | 12, 16 | syl 17 | . . 3 ⊢ (𝑅 ∈ Grp → ((𝐼:(Base‘𝑅)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) ↔ (invg‘𝑂) = 𝐼)) |
18 | 3, 11, 17 | mpbi2and 705 | . 2 ⊢ (𝑅 ∈ Grp → (invg‘𝑂) = 𝐼) |
19 | 18 | eqcomd 2830 | 1 ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3116 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 +gcplusg 16304 0gc0g 16452 Grpcgrp 17775 invgcminusg 17776 oppgcoppg 18124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-tpos 7616 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-plusg 16317 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-grp 17778 df-minusg 17779 df-oppg 18125 |
This theorem is referenced by: oppgsubg 18142 oppgtgp 22271 tgpconncomp 22285 |
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