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Mirrors > Home > MPE Home > Th. List > grpinvval2 | Structured version Visualization version GIF version |
Description: A df-neg 10861-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
grpsubcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubcl.m | ⊢ − = (-g‘𝐺) |
grpinvsub.n | ⊢ 𝑁 = (invg‘𝐺) |
grpinvval2.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpinvval2 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpinvval2.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | 1, 2 | grpidcl 18069 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
4 | eqid 2818 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | grpinvsub.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
6 | grpsubcl.m | . . . 4 ⊢ − = (-g‘𝐺) | |
7 | 1, 4, 5, 6 | grpsubval 18087 | . . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
8 | 3, 7 | sylan 580 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
9 | 1, 5 | grpinvcl 18089 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 1, 4, 2 | grplid 18071 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
11 | 9, 10 | syldan 591 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
12 | 8, 11 | eqtr2d 2854 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Grpcgrp 18041 invgcminusg 18042 -gcsg 18043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 |
This theorem is referenced by: grpsubadd0sub 18124 matinvgcell 20972 istgp2 22627 nrmmetd 23111 nminv 23157 |
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