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Mirrors > Home > ILE Home > Th. List > rngnegr | GIF version |
Description: Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
Ref | Expression |
---|---|
ringnegl.b | ⊢ 𝐵 = (Base‘𝑅) |
ringnegl.t | ⊢ · = (.r‘𝑅) |
ringnegl.u | ⊢ 1 = (1r‘𝑅) |
ringnegl.n | ⊢ 𝑁 = (invg‘𝑅) |
ringnegl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringnegl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
rngnegr | ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) = (𝑁‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringnegl.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringnegl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ringgrp 12997 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
4 | 1, 3 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
5 | ringnegl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
6 | ringnegl.u | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
7 | 5, 6 | ringidcl 13016 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
8 | 1, 7 | syl 14 | . . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐵) |
9 | ringnegl.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
10 | 5, 9 | grpinvcl 12798 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝑁‘ 1 ) ∈ 𝐵) |
11 | 4, 8, 10 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → (𝑁‘ 1 ) ∈ 𝐵) |
12 | eqid 2177 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
13 | ringnegl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
14 | 5, 12, 13 | ringdi 13014 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘ 1 ) ∈ 𝐵 ∧ 1 ∈ 𝐵)) → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 ))) |
15 | 1, 2, 11, 8, 14 | syl13anc 1240 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 ))) |
16 | eqid 2177 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 5, 12, 16, 9 | grplinv 12799 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → ((𝑁‘ 1 )(+g‘𝑅) 1 ) = (0g‘𝑅)) |
18 | 4, 8, 17 | syl2anc 411 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘ 1 )(+g‘𝑅) 1 ) = (0g‘𝑅)) |
19 | 18 | oveq2d 5884 | . . . . 5 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (𝑋 · (0g‘𝑅))) |
20 | 5, 13, 16 | ringrz 13036 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · (0g‘𝑅)) = (0g‘𝑅)) |
21 | 1, 2, 20 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → (𝑋 · (0g‘𝑅)) = (0g‘𝑅)) |
22 | 19, 21 | eqtrd 2210 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (0g‘𝑅)) |
23 | 5, 13, 6 | ringridm 13020 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
24 | 1, 2, 23 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → (𝑋 · 1 ) = 𝑋) |
25 | 24 | oveq2d 5884 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋)) |
26 | 15, 22, 25 | 3eqtr3rd 2219 | . . 3 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅)) |
27 | 5, 13 | ringcl 13009 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘ 1 ) ∈ 𝐵) → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
28 | 1, 2, 11, 27 | syl3anc 1238 | . . . 4 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
29 | 5, 12, 16, 9 | grpinvid2 12802 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) → ((𝑁‘𝑋) = (𝑋 · (𝑁‘ 1 )) ↔ ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅))) |
30 | 4, 2, 28, 29 | syl3anc 1238 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑋 · (𝑁‘ 1 )) ↔ ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅))) |
31 | 26, 30 | mpbird 167 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = (𝑋 · (𝑁‘ 1 ))) |
32 | 31 | eqcomd 2183 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) = (𝑁‘𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 .rcmulr 12506 0gc0g 12640 Grpcgrp 12754 invgcminusg 12755 1rcur 12955 Ringcrg 12992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-inn 8896 df-2 8954 df-3 8955 df-ndx 12435 df-slot 12436 df-base 12438 df-sets 12439 df-plusg 12518 df-mulr 12519 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 df-minusg 12758 df-mgp 12945 df-ur 12956 df-ring 12994 |
This theorem is referenced by: ringmneg2 13044 |
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