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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 35213 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 19312 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐵) |
6 | ringgrp 19296 | . . . . . . 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 18145 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝑁‘ 1 ) ∈ 𝐵) |
10 | 7, 5, 9 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑁‘ 1 ) ∈ 𝐵) |
11 | ringnegl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
13 | ringnegl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
14 | 2, 12, 13 | ringdir 19311 | . . . . 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 1368 | . . . 4 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = (( 1 · 𝑋)(+g‘𝑅)((𝑁‘ 1 ) · 𝑋))) |
16 | eqid 2821 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 2, 12, 16, 8 | grprinv 18147 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → ( 1 (+g‘𝑅)(𝑁‘ 1 )) = (0g‘𝑅)) |
18 | 7, 5, 17 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ( 1 (+g‘𝑅)(𝑁‘ 1 )) = (0g‘𝑅)) |
19 | 18 | oveq1d 7165 | . . . . 5 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = ((0g‘𝑅) · 𝑋)) |
20 | 2, 13, 16 | ringlz 19331 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((0g‘𝑅) · 𝑋) = (0g‘𝑅)) |
21 | 1, 11, 20 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((0g‘𝑅) · 𝑋) = (0g‘𝑅)) |
22 | 19, 21 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = (0g‘𝑅)) |
23 | 2, 13, 3 | ringlidm 19315 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋) |
24 | 1, 11, 23 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
25 | 24 | oveq1d 7165 | . . . 4 ⊢ (𝜑 → (( 1 · 𝑋)(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋))) |
26 | 15, 22, 25 | 3eqtr3rd 2865 | . . 3 ⊢ (𝜑 → (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (0g‘𝑅)) |
27 | 2, 13 | ringcl 19305 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘ 1 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑁‘ 1 ) · 𝑋) ∈ 𝐵) |
28 | 1, 10, 11, 27 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝑋) ∈ 𝐵) |
29 | 2, 12, 16, 8 | grpinvid1 18148 | . . . 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 1367 | . . 3 ⊢ (𝜑 → ((𝑁‘𝑋) = ((𝑁‘ 1 ) · 𝑋) ↔ (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (0g‘𝑅))) |
31 | 26, 30 | mpbird 259 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = ((𝑁‘ 1 ) · 𝑋)) |
32 | 31 | eqcomd 2827 | 1 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝑋) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 0gc0g 16707 Grpcgrp 18097 invgcminusg 18098 1rcur 19245 Ringcrg 19291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-mgp 19234 df-ur 19246 df-ring 19293 |
This theorem is referenced by: ringmneg1 19340 dvdsrneg 19398 abvneg 19599 lmodvsneg 19672 lmodsubvs 19684 lmodsubdi 19685 lmodsubdir 19686 lmodvsinv 19802 mplind 20276 mdetralt 21211 m2detleiblem7 21230 lflsub 36197 baerlem3lem1 38837 |
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