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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 2726 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | irrednu.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | irredn0.i | . . 3 ⊢ 𝐼 = (Irred‘𝑅) | |
4 | eqid 2726 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isirred2 20399 | . 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 1143 | 1 ⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 .rcmulr 17262 Unitcui 20333 Irredcir 20334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-irred 20337 |
This theorem is referenced by: irredn1 20404 mxidlirred 33353 rprmirredb 33413 irredminply 33589 |
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