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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 2736 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | irredn0.i | . . 3 ⊢ 𝐼 = (Irred‘𝑅) | |
4 | eqid 2736 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isirred2 19673 | . 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 1147 | 1 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 .rcmulr 16750 Unitcui 19611 Irredcir 19612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-irred 19615 |
This theorem is referenced by: irredrmul 19679 irredneg 19682 prmirred 20415 |
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