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Mirrors > Home > MPE Home > Th. List > ablinvadd | Structured version Visualization version GIF version |
Description: The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
ablinvadd.b | ⊢ 𝐵 = (Base‘𝐺) |
ablinvadd.p | ⊢ + = (+g‘𝐺) |
ablinvadd.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
ablinvadd | ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑋) + (𝑁‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablgrp 18905 | . . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
2 | ablinvadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | ablinvadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | ablinvadd.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
5 | 2, 3, 4 | grpinvadd 18171 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑁‘𝑋))) |
6 | 1, 5 | syl3an1 1159 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑁‘𝑋))) |
7 | simp1 1132 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Abel) | |
8 | 1 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Grp) |
9 | simp2 1133 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | 2, 4 | grpinvcl 18145 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
11 | 8, 9, 10 | syl2anc 586 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
12 | simp3 1134 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
13 | 2, 4 | grpinvcl 18145 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
14 | 8, 12, 13 | syl2anc 586 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
15 | 2, 3 | ablcom 18918 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ (𝑁‘𝑌) ∈ 𝐵) → ((𝑁‘𝑋) + (𝑁‘𝑌)) = ((𝑁‘𝑌) + (𝑁‘𝑋))) |
16 | 7, 11, 14, 15 | syl3anc 1367 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) + (𝑁‘𝑌)) = ((𝑁‘𝑌) + (𝑁‘𝑋))) |
17 | 6, 16 | eqtr4d 2859 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑋) + (𝑁‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Grpcgrp 18097 invgcminusg 18098 Abelcabl 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-riota 7108 df-ov 7153 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-cmn 18902 df-abl 18903 |
This theorem is referenced by: ablsub4 18927 mulgdi 18941 invghm 18948 lmodnegadd 19677 lflnegcl 36205 baerlem3lem1 38837 |
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