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Mirrors > Home > MPE Home > Th. List > grpinvval2 | Structured version Visualization version GIF version |
Description: A df-neg 11475-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 18924 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
4 | eqid 2725 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | grpinvsub.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
6 | grpsubcl.m | . . . 4 ⊢ − = (-g‘𝐺) | |
7 | 1, 4, 5, 6 | grpsubval 18944 | . . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
8 | 3, 7 | sylan 578 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
9 | 1, 5 | grpinvcl 18946 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 1, 4, 2 | grplid 18926 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
11 | 9, 10 | syldan 589 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
12 | 8, 11 | eqtr2d 2766 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6542 (class class class)co 7415 Basecbs 17177 +gcplusg 17230 0gc0g 17418 Grpcgrp 18892 invgcminusg 18893 -gcsg 18894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 |
This theorem is referenced by: grpsubadd0sub 18985 odm1inv 19510 matinvgcell 22353 istgp2 24011 nrmmetd 24499 nminv 24546 |
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