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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 32911 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 18375 | . . . . . . 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 18391 | . . . . . . 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 17290 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝑁‘ 1 ) ∈ 𝐵) |
11 | 4, 8, 10 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑁‘ 1 ) ∈ 𝐵) |
12 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
13 | ringnegl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
14 | 5, 12, 13 | ringdi 18389 | . . . . 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 1320 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 ))) |
16 | eqid 2610 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 5, 12, 16, 9 | grplinv 17291 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → ((𝑁‘ 1 )(+g‘𝑅) 1 ) = (0g‘𝑅)) |
18 | 4, 8, 17 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘ 1 )(+g‘𝑅) 1 ) = (0g‘𝑅)) |
19 | 18 | oveq2d 6565 | . . . . 5 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (𝑋 · (0g‘𝑅))) |
20 | 5, 13, 16 | ringrz 18411 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · (0g‘𝑅)) = (0g‘𝑅)) |
21 | 1, 2, 20 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑋 · (0g‘𝑅)) = (0g‘𝑅)) |
22 | 19, 21 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → (𝑋 · ((𝑁‘ 1 )(+g‘𝑅) 1 )) = (0g‘𝑅)) |
23 | 5, 13, 6 | ringridm 18395 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
24 | 1, 2, 23 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑋 · 1 ) = 𝑋) |
25 | 24 | oveq2d 6565 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)(𝑋 · 1 )) = ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋)) |
26 | 15, 22, 25 | 3eqtr3rd 2653 | . . 3 ⊢ (𝜑 → ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅)) |
27 | 5, 13 | ringcl 18384 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘ 1 ) ∈ 𝐵) → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
28 | 1, 2, 11, 27 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) ∈ 𝐵) |
29 | 5, 12, 16, 9 | grpinvid2 17294 | . . . 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 1318 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑋 · (𝑁‘ 1 )) ↔ ((𝑋 · (𝑁‘ 1 ))(+g‘𝑅)𝑋) = (0g‘𝑅))) |
31 | 26, 30 | mpbird 246 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = (𝑋 · (𝑁‘ 1 ))) |
32 | 31 | eqcomd 2616 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 1rcur 18324 Ringcrg 18370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 |
This theorem is referenced by: ringmneg2 18420 irredneg 18533 lmodsubdi 18743 mdetunilem7 20243 ldualvsubval 33462 lcdvsubval 35925 mapdpglem30 36009 |
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