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Mirrors > Home > MPE Home > Th. List > grpinvval2 | Structured version Visualization version GIF version |
Description: A df-neg 10611-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 17848 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
4 | eqid 2778 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | grpinvsub.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
6 | grpsubcl.m | . . . 4 ⊢ − = (-g‘𝐺) | |
7 | 1, 4, 5, 6 | grpsubval 17863 | . . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
8 | 3, 7 | sylan 575 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
9 | 1, 5 | grpinvcl 17865 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 1, 4, 2 | grplid 17850 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
11 | 9, 10 | syldan 585 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
12 | 8, 11 | eqtr2d 2815 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 +gcplusg 16349 0gc0g 16497 Grpcgrp 17820 invgcminusg 17821 -gcsg 17822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-0g 16499 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-minusg 17824 df-sbg 17825 |
This theorem is referenced by: grpsubadd0sub 17900 matinvgcell 20656 istgp2 22314 nrmmetd 22798 nminv 22844 |
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