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Theorem isnirred 20420
Description: The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irred.1 𝐵 = (Base‘𝑅)
irred.2 𝑈 = (Unit‘𝑅)
irred.3 𝐼 = (Irred‘𝑅)
irred.4 𝑁 = (𝐵𝑈)
irred.5 · = (.r𝑅)
Assertion
Ref Expression
isnirred (𝑋𝐵 → (¬ 𝑋𝐼 ↔ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)))
Distinct variable groups:   𝑥,𝑦,𝑁   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   · (𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐼(𝑥,𝑦)

Proof of Theorem isnirred
StepHypRef Expression
1 irred.4 . . . . . . 7 𝑁 = (𝐵𝑈)
21eleq2i 2833 . . . . . 6 (𝑋𝑁𝑋 ∈ (𝐵𝑈))
3 eldif 3961 . . . . . 6 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
42, 3bitri 275 . . . . 5 (𝑋𝑁 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
54baibr 536 . . . 4 (𝑋𝐵 → (¬ 𝑋𝑈𝑋𝑁))
6 df-ne 2941 . . . . . . . . 9 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
76ralbii 3093 . . . . . . . 8 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8 ralnex 3072 . . . . . . . 8 (∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
97, 8bitri 275 . . . . . . 7 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
109ralbii 3093 . . . . . 6 (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
11 ralnex 3072 . . . . . 6 (∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
1210, 11bitr2i 276 . . . . 5 (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)
1312a1i 11 . . . 4 (𝑋𝐵 → (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
145, 13anbi12d 632 . . 3 (𝑋𝐵 → ((¬ 𝑋𝑈 ∧ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15 ioran 986 . . 3 (¬ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (¬ 𝑋𝑈 ∧ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋))
16 irred.1 . . . 4 𝐵 = (Base‘𝑅)
17 irred.2 . . . 4 𝑈 = (Unit‘𝑅)
18 irred.3 . . . 4 𝐼 = (Irred‘𝑅)
19 irred.5 . . . 4 · = (.r𝑅)
2016, 17, 18, 1, 19isirred 20419 . . 3 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
2114, 15, 203bitr4g 314 . 2 (𝑋𝐵 → (¬ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ 𝑋𝐼))
2221con1bid 355 1 (𝑋𝐵 → (¬ 𝑋𝐼 ↔ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cdif 3948  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  Unitcui 20355  Irredcir 20356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-irred 20359
This theorem is referenced by:  irredn0  20423  irredrmul  20427
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