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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 2735 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | irredn0.i | . . 3 ⊢ 𝐼 = (Irred‘𝑅) | |
4 | eqid 2735 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isirred2 20438 | . 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 1144 | 1 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 Unitcui 20372 Irredcir 20373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-irred 20376 |
This theorem is referenced by: irredrmul 20444 irredneg 20447 prmirred 21503 irrednzr 33237 rprmirredb 33540 irredminply 33722 |
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