![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 35379 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 19314 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐵) |
6 | ringgrp 19295 | . . . . . . 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 18143 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → (𝑁‘ 1 ) ∈ 𝐵) |
10 | 7, 5, 9 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑁‘ 1 ) ∈ 𝐵) |
11 | ringnegl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | eqid 2798 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
13 | ringnegl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
14 | 2, 12, 13 | ringdir 19313 | . . . . 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 2798 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | 2, 12, 16, 8 | grprinv 18145 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → ( 1 (+g‘𝑅)(𝑁‘ 1 )) = (0g‘𝑅)) |
18 | 7, 5, 17 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ( 1 (+g‘𝑅)(𝑁‘ 1 )) = (0g‘𝑅)) |
19 | 18 | oveq1d 7150 | . . . . 5 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = ((0g‘𝑅) · 𝑋)) |
20 | 2, 13, 16 | ringlz 19333 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((0g‘𝑅) · 𝑋) = (0g‘𝑅)) |
21 | 1, 11, 20 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ((0g‘𝑅) · 𝑋) = (0g‘𝑅)) |
22 | 19, 21 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → (( 1 (+g‘𝑅)(𝑁‘ 1 )) · 𝑋) = (0g‘𝑅)) |
23 | 2, 13, 3 | ringlidm 19317 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋) |
24 | 1, 11, 23 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
25 | 24 | oveq1d 7150 | . . . 4 ⊢ (𝜑 → (( 1 · 𝑋)(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋))) |
26 | 15, 22, 25 | 3eqtr3rd 2842 | . . 3 ⊢ (𝜑 → (𝑋(+g‘𝑅)((𝑁‘ 1 ) · 𝑋)) = (0g‘𝑅)) |
27 | 2, 13 | ringcl 19307 | . . . . 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 18146 | . . . 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 260 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = ((𝑁‘ 1 ) · 𝑋)) |
32 | 31 | eqcomd 2804 | 1 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝑋) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 1rcur 19244 Ringcrg 19290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mgp 19233 df-ur 19245 df-ring 19292 |
This theorem is referenced by: ringmneg1 19342 dvdsrneg 19400 abvneg 19598 lmodvsneg 19671 lmodsubvs 19683 lmodsubdi 19684 lmodsubdir 19685 lmodvsinv 19801 mplind 20741 mdetralt 21213 m2detleiblem7 21232 lflsub 36363 baerlem3lem1 39003 |
Copyright terms: Public domain | W3C validator |