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Mirrors > Home > MPE Home > Th. List > ringnegl | Structured version Visualization version GIF version |
Description: Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 37642 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 |
---|---|
ringnegl | ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝑋) = (𝑁‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringnegl.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringnegl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
3 | ringnegl.u | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 20245 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐵) |
6 | ringgrp 20221 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 1, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
8 | ringnegl.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
9 | 2, 8 | grpinvcl 18982 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝑁‘ 1 ) ∈ 𝐵) |
10 | 7, 5, 9 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝑁‘ 1 ) ∈ 𝐵) |
11 | ringnegl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | eqid 2726 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
13 | ringnegl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
14 | 2, 12, 13 | ringdir 20244 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ( 1 ∈ 𝐵 ∧ (𝑁‘ 1 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = (( 1 · 𝑋)(+g‘𝑅)((𝑁‘ 1 ) · 𝑋))) |
15 | 1, 5, 10, 11, 14 | syl13anc 1369 | . . . 4 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = (( 1 · 𝑋)(+g‘𝑅)((𝑁‘ 1 ) · 𝑋))) |
16 | eqid 2726 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 2, 12, 16, 8 | grprinv 18985 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → ( 1 (+g‘𝑅)(𝑁‘ 1 )) = (0g‘𝑅)) |
18 | 7, 5, 17 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → ( 1 (+g‘𝑅)(𝑁‘ 1 )) = (0g‘𝑅)) |
19 | 18 | oveq1d 7439 | . . . . 5 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = ((0g‘𝑅) · 𝑋)) |
20 | 2, 13, 16 | ringlz 20272 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((0g‘𝑅) · 𝑋) = (0g‘𝑅)) |
21 | 1, 11, 20 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ((0g‘𝑅) · 𝑋) = (0g‘𝑅)) |
22 | 19, 21 | eqtrd 2766 | . . . 4 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = (0g‘𝑅)) |
23 | 2, 13, 3 | ringlidm 20248 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋) |
24 | 1, 11, 23 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
25 | 24 | oveq1d 7439 | . . . 4 ⊢ (𝜑 → (( 1 · 𝑋)(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋))) |
26 | 15, 22, 25 | 3eqtr3rd 2775 | . . 3 ⊢ (𝜑 → (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (0g‘𝑅)) |
27 | 2, 13 | ringcl 20233 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘ 1 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑁‘ 1 ) · 𝑋) ∈ 𝐵) |
28 | 1, 10, 11, 27 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝑋) ∈ 𝐵) |
29 | 2, 12, 16, 8 | grpinvid1 18986 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ((𝑁‘ 1 ) · 𝑋) ∈ 𝐵) → ((𝑁‘𝑋) = ((𝑁‘ 1 ) · 𝑋) ↔ (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (0g‘𝑅))) |
30 | 7, 11, 28, 29 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋) = ((𝑁‘ 1 ) · 𝑋) ↔ (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (0g‘𝑅))) |
31 | 26, 30 | mpbird 256 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = ((𝑁‘ 1 ) · 𝑋)) |
32 | 31 | eqcomd 2732 | 1 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝑋) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 .rcmulr 17267 0gc0g 17454 Grpcgrp 18928 invgcminusg 18929 1rcur 20164 Ringcrg 20216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 |
This theorem is referenced by: ringmneg1 20283 dvdsrneg 20352 abvneg 20805 lmodvsneg 20882 lmodsubvs 20894 lmodsubdi 20895 lmodsubdir 20896 lmodvsinv 21014 mplind 22083 mdetralt 22601 m2detleiblem7 22620 lflsub 38765 baerlem3lem1 41406 |
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