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Mirrors > Home > MPE Home > Th. List > grpinvval2 | Structured version Visualization version GIF version |
Description: A df-neg 10148-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 17273 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
4 | eqid 2610 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | grpinvsub.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
6 | grpsubcl.m | . . . 4 ⊢ − = (-g‘𝐺) | |
7 | 1, 4, 5, 6 | grpsubval 17288 | . . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
8 | 3, 7 | sylan 487 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 − 𝑋) = ( 0 (+g‘𝐺)(𝑁‘𝑋))) |
9 | 1, 5 | grpinvcl 17290 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
10 | 1, 4, 2 | grplid 17275 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
11 | 9, 10 | syldan 486 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝐺)(𝑁‘𝑋)) = (𝑁‘𝑋)) |
12 | 8, 11 | eqtr2d 2645 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 -gcsg 17247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 |
This theorem is referenced by: grpsubadd0sub 17325 matinvgcell 20060 istgp2 21705 nrmmetd 22189 nminv 22235 |
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