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Mirrors > Home > MPE Home > Th. List > grpsubinv | Structured version Visualization version GIF version |
Description: Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.) |
Ref | Expression |
---|---|
grpsubinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubinv.p | ⊢ + = (+g‘𝐺) |
grpsubinv.m | ⊢ − = (-g‘𝐺) |
grpsubinv.n | ⊢ 𝑁 = (invg‘𝐺) |
grpsubinv.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
grpsubinv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
grpsubinv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
grpsubinv | ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubinv.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | grpsubinv.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
3 | grpsubinv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | grpsubinv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | grpsubinv.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 4, 5 | grpinvcl 17780 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
7 | 2, 3, 6 | syl2anc 580 | . . 3 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
8 | grpsubinv.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | grpsubinv.m | . . . 4 ⊢ − = (-g‘𝐺) | |
10 | 4, 8, 5, 9 | grpsubval 17778 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑁‘𝑌) ∈ 𝐵) → (𝑋 − (𝑁‘𝑌)) = (𝑋 + (𝑁‘(𝑁‘𝑌)))) |
11 | 1, 7, 10 | syl2anc 580 | . 2 ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + (𝑁‘(𝑁‘𝑌)))) |
12 | 4, 5 | grpinvinv 17795 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
13 | 2, 3, 12 | syl2anc 580 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
14 | 13 | oveq2d 6892 | . 2 ⊢ (𝜑 → (𝑋 + (𝑁‘(𝑁‘𝑌))) = (𝑋 + 𝑌)) |
15 | 11, 14 | eqtrd 2831 | 1 ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 Basecbs 16181 +gcplusg 16264 Grpcgrp 17735 invgcminusg 17736 -gcsg 17737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-1st 7399 df-2nd 7400 df-0g 16414 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-minusg 17739 df-sbg 17740 |
This theorem is referenced by: issubg4 17923 isnsg3 17938 lsmelvalm 18376 ablsub2inv 18528 ablsubsub4 18536 istgp2 22220 nmtri 22755 baerlem5amN 37729 baerlem5abmN 37731 |
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