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Mirrors > Home > MPE Home > Th. List > rngnegr | Structured version Visualization version GIF version |
Description: Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 35222 analog.) (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 19304 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
5 | ringnegl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
6 | ringnegl.u | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
7 | 5, 6 | ringidcl 19320 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐵) |
9 | ringnegl.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
10 | 5, 9 | grpinvcl 18153 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝑁‘ 1 ) ∈ 𝐵) |
11 | 4, 8, 10 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑁‘ 1 ) ∈ 𝐵) |
12 | eqid 2823 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
13 | ringnegl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
14 | 5, 12, 13 | ringdi 19318 | . . . . 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 1368 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 ))) |
16 | eqid 2823 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 5, 12, 16, 9 | grplinv 18154 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → ((𝑁‘ 1 )(+g‘𝑅) 1 ) = (0g‘𝑅)) |
18 | 4, 8, 17 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘ 1 )(+g‘𝑅) 1 ) = (0g‘𝑅)) |
19 | 18 | oveq2d 7174 | . . . . 5 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (𝑋 · (0g‘𝑅))) |
20 | 5, 13, 16 | ringrz 19340 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · (0g‘𝑅)) = (0g‘𝑅)) |
21 | 1, 2, 20 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑋 · (0g‘𝑅)) = (0g‘𝑅)) |
22 | 19, 21 | eqtrd 2858 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (0g‘𝑅)) |
23 | 5, 13, 6 | ringridm 19324 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
24 | 1, 2, 23 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑋 · 1 ) = 𝑋) |
25 | 24 | oveq2d 7174 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋)) |
26 | 15, 22, 25 | 3eqtr3rd 2867 | . . 3 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅)) |
27 | 5, 13 | ringcl 19313 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘ 1 ) ∈ 𝐵) → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
28 | 1, 2, 11, 27 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
29 | 5, 12, 16, 9 | grpinvid2 18157 | . . . 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 1367 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑋 · (𝑁‘ 1 )) ↔ ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅))) |
31 | 26, 30 | mpbird 259 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = (𝑋 · (𝑁‘ 1 ))) |
32 | 31 | eqcomd 2829 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 0gc0g 16715 Grpcgrp 18105 invgcminusg 18106 1rcur 19253 Ringcrg 19299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-mgp 19242 df-ur 19254 df-ring 19301 |
This theorem is referenced by: ringmneg2 19349 irredneg 19462 lmodsubdi 19693 mdetunilem7 21229 ldualvsubval 36295 lcdvsubval 38756 mapdpglem30 38840 |
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