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Mirrors > Home > MPE Home > Th. List > irredcl | Structured version Visualization version GIF version |
Description: An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
irredn0.i | ⊢ 𝐼 = (Irred‘𝑅) |
irredcl.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
irredcl | ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | irredcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2818 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | irredn0.i | . . 3 ⊢ 𝐼 = (Irred‘𝑅) | |
4 | eqid 2818 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isirred2 19380 | . 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ (Unit‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑋 → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅))))) |
6 | 5 | simp1bi 1137 | 1 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 .rcmulr 16554 Unitcui 19318 Irredcir 19319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-irred 19322 |
This theorem is referenced by: irredrmul 19386 irredneg 19389 prmirred 20570 |
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