![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > irredn0 | Structured version Visualization version GIF version |
Description: The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
irredn0.i | ⊢ 𝐼 = (Irred‘𝑅) |
irredn0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
irredn0 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2760 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | irredn0.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | ring0cl 18789 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
4 | 3 | anim1i 593 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → ( 0 ∈ (Base‘𝑅) ∧ ¬ 0 ∈ (Unit‘𝑅))) |
5 | eldif 3725 | . . . . . . . 8 ⊢ ( 0 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ↔ ( 0 ∈ (Base‘𝑅) ∧ ¬ 0 ∈ (Unit‘𝑅))) | |
6 | 4, 5 | sylibr 224 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → 0 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) |
7 | eqid 2760 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
8 | 1, 7, 2 | ringlz 18807 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ (Base‘𝑅)) → ( 0 (.r‘𝑅) 0 ) = 0 ) |
9 | 3, 8 | mpdan 705 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ( 0 (.r‘𝑅) 0 ) = 0 ) |
10 | 9 | adantr 472 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → ( 0 (.r‘𝑅) 0 ) = 0 ) |
11 | oveq1 6821 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥(.r‘𝑅)𝑦) = ( 0 (.r‘𝑅)𝑦)) | |
12 | 11 | eqeq1d 2762 | . . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥(.r‘𝑅)𝑦) = 0 ↔ ( 0 (.r‘𝑅)𝑦) = 0 )) |
13 | oveq2 6822 | . . . . . . . . 9 ⊢ (𝑦 = 0 → ( 0 (.r‘𝑅)𝑦) = ( 0 (.r‘𝑅) 0 )) | |
14 | 13 | eqeq1d 2762 | . . . . . . . 8 ⊢ (𝑦 = 0 → (( 0 (.r‘𝑅)𝑦) = 0 ↔ ( 0 (.r‘𝑅) 0 ) = 0 )) |
15 | 12, 14 | rspc2ev 3463 | . . . . . . 7 ⊢ (( 0 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ∧ 0 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ∧ ( 0 (.r‘𝑅) 0 ) = 0 ) → ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 ) |
16 | 6, 6, 10, 15 | syl3anc 1477 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 ) |
17 | 16 | ex 449 | . . . . 5 ⊢ (𝑅 ∈ Ring → (¬ 0 ∈ (Unit‘𝑅) → ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 )) |
18 | 17 | orrd 392 | . . . 4 ⊢ (𝑅 ∈ Ring → ( 0 ∈ (Unit‘𝑅) ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 )) |
19 | eqid 2760 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
20 | irredn0.i | . . . . . 6 ⊢ 𝐼 = (Irred‘𝑅) | |
21 | eqid 2760 | . . . . . 6 ⊢ ((Base‘𝑅) ∖ (Unit‘𝑅)) = ((Base‘𝑅) ∖ (Unit‘𝑅)) | |
22 | 1, 19, 20, 21, 7 | isnirred 18920 | . . . . 5 ⊢ ( 0 ∈ (Base‘𝑅) → (¬ 0 ∈ 𝐼 ↔ ( 0 ∈ (Unit‘𝑅) ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 ))) |
23 | 3, 22 | syl 17 | . . . 4 ⊢ (𝑅 ∈ Ring → (¬ 0 ∈ 𝐼 ↔ ( 0 ∈ (Unit‘𝑅) ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 ))) |
24 | 18, 23 | mpbird 247 | . . 3 ⊢ (𝑅 ∈ Ring → ¬ 0 ∈ 𝐼) |
25 | 24 | adantr 472 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → ¬ 0 ∈ 𝐼) |
26 | simpr 479 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ∈ 𝐼) | |
27 | eleq1 2827 | . . . 4 ⊢ (𝑋 = 0 → (𝑋 ∈ 𝐼 ↔ 0 ∈ 𝐼)) | |
28 | 26, 27 | syl5ibcom 235 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → (𝑋 = 0 → 0 ∈ 𝐼)) |
29 | 28 | necon3bd 2946 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → (¬ 0 ∈ 𝐼 → 𝑋 ≠ 0 )) |
30 | 25, 29 | mpd 15 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∃wrex 3051 ∖ cdif 3712 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 .rcmulr 16164 0gc0g 16322 Ringcrg 18767 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-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-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-plusg 16176 df-0g 16324 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-minusg 17647 df-mgp 18710 df-ring 18769 df-irred 18863 |
This theorem is referenced by: prmirred 20065 |
Copyright terms: Public domain | W3C validator |