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Mirrors > Home > MPE Home > Th. List > irrednu | Structured version Visualization version GIF version |
Description: An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
irredn0.i | ⊢ 𝐼 = (Irred‘𝑅) |
irrednu.u | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
irrednu | ⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | irrednu.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | irredn0.i | . . 3 ⊢ 𝐼 = (Irred‘𝑅) | |
4 | eqid 2622 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isirred2 18701 | . 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))) |
6 | 5 | simp2bi 1077 | 1 ⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 Unitcui 18639 Irredcir 18640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-irred 18643 |
This theorem is referenced by: irredn1 18706 |
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