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Mirrors > Home > MPE Home > Th. List > ringmneg2 | Structured version Visualization version GIF version |
Description: Negation of a product in a ring. (mulneg2 10812 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
Ref | Expression |
---|---|
ringneglmul.b | ⊢ 𝐵 = (Base‘𝑅) |
ringneglmul.t | ⊢ · = (.r‘𝑅) |
ringneglmul.n | ⊢ 𝑁 = (invg‘𝑅) |
ringneglmul.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringneglmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringneglmul.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
ringmneg2 | ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneglmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringneglmul.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ringneglmul.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ringgrp 18939 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
6 | ringneglmul.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
7 | eqid 2777 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 6, 7 | ringidcl 18955 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
10 | ringneglmul.n | . . . . 5 ⊢ 𝑁 = (invg‘𝑅) | |
11 | 6, 10 | grpinvcl 17854 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ 𝐵) → (𝑁‘(1r‘𝑅)) ∈ 𝐵) |
12 | 5, 9, 11 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝑁‘(1r‘𝑅)) ∈ 𝐵) |
13 | ringneglmul.t | . . . 4 ⊢ · = (.r‘𝑅) | |
14 | 6, 13 | ringass 18951 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑁‘(1r‘𝑅)) ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅))))) |
15 | 1, 2, 3, 12, 14 | syl13anc 1440 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅))))) |
16 | 6, 13 | ringcl 18948 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
17 | 1, 2, 3, 16 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
18 | 6, 13, 7, 10, 1, 17 | rngnegr 18982 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑁‘(𝑋 · 𝑌))) |
19 | 6, 13, 7, 10, 1, 3 | rngnegr 18982 | . . 3 ⊢ (𝜑 → (𝑌 · (𝑁‘(1r‘𝑅))) = (𝑁‘𝑌)) |
20 | 19 | oveq2d 6938 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅)))) = (𝑋 · (𝑁‘𝑌))) |
21 | 15, 18, 20 | 3eqtr3rd 2822 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 .rcmulr 16339 Grpcgrp 17809 invgcminusg 17810 1rcur 18888 Ringcrg 18934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-mgp 18877 df-ur 18889 df-ring 18936 |
This theorem is referenced by: ringm2neg 18985 ringsubdi 18986 cntzsubr 19204 abvneg 19226 lflnegcl 35224 |
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