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| Mirrors > Home > MPE Home > Th. List > grpinvval2 | Structured version Visualization version GIF version | ||
| Description: A df-neg 11495-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 18983 | . . 3 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
| 4 | eqid 2737 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 5 | grpinvsub.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
| 6 | grpsubcl.m | . . . 4 ⊢ − = (-g‘𝐺) | |
| 7 | 1, 4, 5, 6 | grpsubval 19003 | . . 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 19005 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
| 10 | 1, 4, 2 | grplid 18985 | . . 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 2778 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ( 0 − 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 invgcminusg 18952 -gcsg 18953 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 |
| This theorem is referenced by: grpsubadd0sub 19045 odm1inv 19571 matinvgcell 22441 istgp2 24099 nrmmetd 24587 nminv 24634 |
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