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Mirrors > Home > MPE Home > Th. List > grpinvcnv | Structured version Visualization version GIF version |
Description: The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
grpinvinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvinv.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvcnv | ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) | |
2 | grpinvinv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpinvinv.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
4 | 2, 3 | grpinvcl 18145 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑁‘𝑥) ∈ 𝐵) |
5 | 2, 3 | grpinvcl 18145 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝑁‘𝑦) ∈ 𝐵) |
6 | eqid 2821 | . . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | eqid 2821 | . . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 2, 6, 7, 3 | grpinvid1 18148 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
9 | 8 | 3com23 1122 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
10 | 2, 6, 7, 3 | grpinvid2 18149 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑥) = 𝑦 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
11 | 9, 10 | bitr4d 284 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦)) |
12 | 11 | 3expb 1116 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦)) |
13 | eqcom 2828 | . . . . 5 ⊢ (𝑥 = (𝑁‘𝑦) ↔ (𝑁‘𝑦) = 𝑥) | |
14 | eqcom 2828 | . . . . 5 ⊢ (𝑦 = (𝑁‘𝑥) ↔ (𝑁‘𝑥) = 𝑦) | |
15 | 12, 13, 14 | 3bitr4g 316 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝑁‘𝑦) ↔ 𝑦 = (𝑁‘𝑥))) |
16 | 1, 4, 5, 15 | f1ocnv2d 7392 | . . 3 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)):𝐵–1-1-onto→𝐵 ∧ ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦)))) |
17 | 16 | simprd 498 | . 2 ⊢ (𝐺 ∈ Grp → ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦))) |
18 | 2, 3 | grpinvf 18144 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) |
19 | 18 | feqmptd 6728 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥))) |
20 | 19 | cnveqd 5741 | . 2 ⊢ (𝐺 ∈ Grp → ◡𝑁 = ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥))) |
21 | 18 | feqmptd 6728 | . 2 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦))) |
22 | 17, 20, 21 | 3eqtr4d 2866 | 1 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5139 ◡ccnv 5549 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Grpcgrp 18097 invgcminusg 18098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 |
This theorem is referenced by: grpinvf1o 18163 grpinvhmeo 22688 |
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