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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 2823 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | irredn0.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | ring0cl 19321 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
4 | 3 | anim1i 616 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → ( 0 ∈ (Base‘𝑅) ∧ ¬ 0 ∈ (Unit‘𝑅))) |
5 | eldif 3948 | . . . . . . . 8 ⊢ ( 0 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ↔ ( 0 ∈ (Base‘𝑅) ∧ ¬ 0 ∈ (Unit‘𝑅))) | |
6 | 4, 5 | sylibr 236 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → 0 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) |
7 | eqid 2823 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
8 | 1, 7, 2 | ringlz 19339 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ (Base‘𝑅)) → ( 0 (.r‘𝑅) 0 ) = 0 ) |
9 | 3, 8 | mpdan 685 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ( 0 (.r‘𝑅) 0 ) = 0 ) |
10 | 9 | adantr 483 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → ( 0 (.r‘𝑅) 0 ) = 0 ) |
11 | oveq1 7165 | . . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥(.r‘𝑅)𝑦) = ( 0 (.r‘𝑅)𝑦)) | |
12 | 11 | eqeq1d 2825 | . . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥(.r‘𝑅)𝑦) = 0 ↔ ( 0 (.r‘𝑅)𝑦) = 0 )) |
13 | oveq2 7166 | . . . . . . . . 9 ⊢ (𝑦 = 0 → ( 0 (.r‘𝑅)𝑦) = ( 0 (.r‘𝑅) 0 )) | |
14 | 13 | eqeq1d 2825 | . . . . . . . 8 ⊢ (𝑦 = 0 → (( 0 (.r‘𝑅)𝑦) = 0 ↔ ( 0 (.r‘𝑅) 0 ) = 0 )) |
15 | 12, 14 | rspc2ev 3637 | . . . . . . 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 1367 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ¬ 0 ∈ (Unit‘𝑅)) → ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 ) |
17 | 16 | ex 415 | . . . . 5 ⊢ (𝑅 ∈ Ring → (¬ 0 ∈ (Unit‘𝑅) → ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 )) |
18 | 17 | orrd 859 | . . . 4 ⊢ (𝑅 ∈ Ring → ( 0 ∈ (Unit‘𝑅) ∨ ∃𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∃𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑥(.r‘𝑅)𝑦) = 0 )) |
19 | eqid 2823 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
20 | irredn0.i | . . . . . 6 ⊢ 𝐼 = (Irred‘𝑅) | |
21 | eqid 2823 | . . . . . 6 ⊢ ((Base‘𝑅) ∖ (Unit‘𝑅)) = ((Base‘𝑅) ∖ (Unit‘𝑅)) | |
22 | 1, 19, 20, 21, 7 | isnirred 19452 | . . . . 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 259 | . . 3 ⊢ (𝑅 ∈ Ring → ¬ 0 ∈ 𝐼) |
25 | 24 | adantr 483 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → ¬ 0 ∈ 𝐼) |
26 | simpr 487 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ∈ 𝐼) | |
27 | eleq1 2902 | . . . 4 ⊢ (𝑋 = 0 → (𝑋 ∈ 𝐼 ↔ 0 ∈ 𝐼)) | |
28 | 26, 27 | syl5ibcom 247 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → (𝑋 = 0 → 0 ∈ 𝐼)) |
29 | 28 | necon3bd 3032 | . 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 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 ∖ cdif 3935 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 0gc0g 16715 Ringcrg 19299 Unitcui 19391 Irredcir 19392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-mgp 19242 df-ring 19301 df-irred 19395 |
This theorem is referenced by: prmirred 20644 |
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