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| Mirrors > Home > ILE Home > Th. List > ringnegr | 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 |
|---|---|
| ringnegr | ⊢ (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 14017 | . . . . . . 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 14036 | . . . . . . 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 13633 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝑁‘ 1 ) ∈ 𝐵) |
| 11 | 4, 8, 10 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → (𝑁‘ 1 ) ∈ 𝐵) |
| 12 | eqid 2231 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 13 | ringnegl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
| 14 | 5, 12, 13 | ringdi 14034 | . . . . 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 1275 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 ))) |
| 16 | eqid 2231 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 17 | 5, 12, 16, 9 | grplinv 13635 | . . . . . . 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 6034 | . . . . 5 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (𝑋 · (0g‘𝑅))) |
| 20 | 5, 13, 16 | ringrz 14060 | . . . . . 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 2264 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (0g‘𝑅)) |
| 23 | 5, 13, 6 | ringridm 14040 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
| 24 | 1, 2, 23 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → (𝑋 · 1 ) = 𝑋) |
| 25 | 24 | oveq2d 6034 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋)) |
| 26 | 15, 22, 25 | 3eqtr3rd 2273 | . . 3 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅)) |
| 27 | 5, 13 | ringcl 14029 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘ 1 ) ∈ 𝐵) → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
| 28 | 1, 2, 11, 27 | syl3anc 1273 | . . . 4 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
| 29 | 5, 12, 16, 9 | grpinvid2 13638 | . . . 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 1273 | . . 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 2237 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) = (𝑁‘𝑋)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ↔ wb 105 = wceq 1397 ∈ wcel 2202 ‘cfv 5326 (class class class)co 6018 Basecbs 13084 +gcplusg 13162 .rcmulr 13163 0gc0g 13341 Grpcgrp 13585 invgcminusg 13586 1rcur 13975 Ringcrg 14012 |
| 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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-addcom 8132 ax-addass 8134 ax-i2m1 8137 ax-0lt1 8138 ax-0id 8140 ax-rnegex 8141 ax-pre-ltirr 8144 ax-pre-ltadd 8148 |
| This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-ltxr 8219 df-inn 9144 df-2 9202 df-3 9203 df-ndx 13087 df-slot 13088 df-base 13090 df-sets 13091 df-plusg 13175 df-mulr 13176 df-0g 13343 df-mgm 13441 df-sgrp 13487 df-mnd 13502 df-grp 13588 df-minusg 13589 df-mgp 13937 df-ur 13976 df-ring 14014 |
| This theorem is referenced by: ringmneg2 14070 lmodsubdi 14361 |
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