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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 2724 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | irredrmul.t | . . 3 ⊢ · = (.r‘𝑅) | |
3 | eqid 2724 | . . 3 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
4 | eqid 2724 | . . 3 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)) | |
5 | 1, 2, 3, 4 | opprmul 20224 | . 2 ⊢ (𝑌(.r‘(oppr‘𝑅))𝑋) = (𝑋 · 𝑌) |
6 | 3 | opprring 20234 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) |
7 | irredn0.i | . . . . . 6 ⊢ 𝐼 = (Irred‘𝑅) | |
8 | 3, 7 | opprirred 20309 | . . . . 5 ⊢ 𝐼 = (Irred‘(oppr‘𝑅)) |
9 | irredrmul.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
10 | 9, 3 | opprunit 20264 | . . . . 5 ⊢ 𝑈 = (Unit‘(oppr‘𝑅)) |
11 | 8, 10, 4 | irredrmul 20314 | . . . 4 ⊢ (((oppr‘𝑅) ∈ Ring ∧ 𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈) → (𝑌(.r‘(oppr‘𝑅))𝑋) ∈ 𝐼) |
12 | 6, 11 | syl3an1 1160 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈) → (𝑌(.r‘(oppr‘𝑅))𝑋) ∈ 𝐼) |
13 | 12 | 3com23 1123 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼) → (𝑌(.r‘(oppr‘𝑅))𝑋) ∈ 𝐼) |
14 | 5, 13 | eqeltrrid 2830 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼) → (𝑋 · 𝑌) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 Basecbs 17140 .rcmulr 17194 Ringcrg 20123 opprcoppr 20220 Unitcui 20242 Irredcir 20243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-grp 18853 df-minusg 18854 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-oppr 20221 df-dvdsr 20244 df-unit 20245 df-irred 20246 df-invr 20275 df-dvr 20288 |
This theorem is referenced by: (None) |
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