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Mirrors > Home > MPE Home > Th. List > grpinvinv | Structured version Visualization version GIF version |
Description: Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
grpinvinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvinv.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvinv | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvinv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpinvinv.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 18154 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
4 | eqid 2824 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | eqid 2824 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
6 | 1, 4, 5, 2 | grprinv 18156 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0g‘𝐺)) |
7 | 3, 6 | syldan 593 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0g‘𝐺)) |
8 | 1, 4, 5, 2 | grplinv 18155 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)𝑋) = (0g‘𝐺)) |
9 | 7, 8 | eqtr4d 2862 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+g‘𝐺)𝑋)) |
10 | simpl 485 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Grp) | |
11 | 1, 2 | grpinvcl 18154 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵) |
12 | 3, 11 | syldan 593 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵) |
13 | simpr 487 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
14 | 1, 4 | grplcan 18164 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ((𝑁‘(𝑁‘𝑋)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵)) → (((𝑁‘𝑋)(+g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+g‘𝐺)𝑋) ↔ (𝑁‘(𝑁‘𝑋)) = 𝑋)) |
15 | 10, 12, 13, 3, 14 | syl13anc 1368 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((𝑁‘𝑋)(+g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+g‘𝐺)𝑋) ↔ (𝑁‘(𝑁‘𝑋)) = 𝑋)) |
16 | 9, 15 | mpbid 234 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 0gc0g 16716 Grpcgrp 18106 invgcminusg 18107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-riota 7117 df-ov 7162 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 |
This theorem is referenced by: grpinv11 18171 grpinvnz 18173 grpsubinv 18175 grpinvsub 18184 grpsubeq0 18188 grpnpcan 18194 mulgneg 18249 mulgnegneg 18250 mulginvinv 18256 mulgdir 18262 mulgass 18267 eqger 18333 frgpuptinv 18900 ablsub2inv 18934 mulgdi 18950 invghm 18957 ringm2neg 19351 unitinvinv 19428 unitnegcl 19434 irrednegb 19464 abvneg 19608 lspsnneg 19781 islindf4 20985 tgpconncomp 22724 archirngz 30822 archiabllem1b 30825 baerlem5amN 38856 baerlem5bmN 38857 baerlem5abmN 38858 nelsubginvcld 39134 |
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