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Mirrors > Home > MPE Home > Th. List > irredlmul | Structured version Visualization version GIF version |
Description: The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
irredn0.i | ⊢ 𝐼 = (Irred‘𝑅) |
irredrmul.u | ⊢ 𝑈 = (Unit‘𝑅) |
irredrmul.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
irredlmul | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼) → (𝑋 · 𝑌) ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2760 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | irredrmul.t | . . 3 ⊢ · = (.r‘𝑅) | |
3 | eqid 2760 | . . 3 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
4 | eqid 2760 | . . 3 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)) | |
5 | 1, 2, 3, 4 | opprmul 18846 | . 2 ⊢ (𝑌(.r‘(oppr‘𝑅))𝑋) = (𝑋 · 𝑌) |
6 | 3 | opprring 18851 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) |
7 | irredn0.i | . . . . . 6 ⊢ 𝐼 = (Irred‘𝑅) | |
8 | 3, 7 | opprirred 18922 | . . . . 5 ⊢ 𝐼 = (Irred‘(oppr‘𝑅)) |
9 | irredrmul.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
10 | 9, 3 | opprunit 18881 | . . . . 5 ⊢ 𝑈 = (Unit‘(oppr‘𝑅)) |
11 | 8, 10, 4 | irredrmul 18927 | . . . 4 ⊢ (((oppr‘𝑅) ∈ Ring ∧ 𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈) → (𝑌(.r‘(oppr‘𝑅))𝑋) ∈ 𝐼) |
12 | 6, 11 | syl3an1 1167 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈) → (𝑌(.r‘(oppr‘𝑅))𝑋) ∈ 𝐼) |
13 | 12 | 3com23 1121 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼) → (𝑌(.r‘(oppr‘𝑅))𝑋) ∈ 𝐼) |
14 | 5, 13 | syl5eqelr 2844 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼) → (𝑋 · 𝑌) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 .rcmulr 16164 Ringcrg 18767 opprcoppr 18842 Unitcui 18859 Irredcir 18860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-0g 16324 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-minusg 17647 df-mgp 18710 df-ur 18722 df-ring 18769 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-irred 18863 df-invr 18892 df-dvr 18903 |
This theorem is referenced by: (None) |
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