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Theorem isnirred 18746
 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 2722 . . . . . 6 (𝑋𝑁𝑋 ∈ (𝐵𝑈))
3 eldif 3617 . . . . . 6 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
42, 3bitri 264 . . . . 5 (𝑋𝑁 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
54baibr 965 . . . 4 (𝑋𝐵 → (¬ 𝑋𝑈𝑋𝑁))
6 df-ne 2824 . . . . . . . . 9 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
76ralbii 3009 . . . . . . . 8 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8 ralnex 3021 . . . . . . . 8 (∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
97, 8bitri 264 . . . . . . 7 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
109ralbii 3009 . . . . . 6 (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
11 ralnex 3021 . . . . . 6 (∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
1210, 11bitr2i 265 . . . . 5 (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)
1312a1i 11 . . . 4 (𝑋𝐵 → (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
145, 13anbi12d 747 . . 3 (𝑋𝐵 → ((¬ 𝑋𝑈 ∧ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15 ioran 510 . . 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 18745 . . 3 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
2114, 15, 203bitr4g 303 . 2 (𝑋𝐵 → (¬ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ 𝑋𝐼))
2221con1bid 344 1 (𝑋𝐵 → (¬ 𝑋𝐼 ↔ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942   ∖ cdif 3604  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989  Unitcui 18685  Irredcir 18686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-irred 18689 This theorem is referenced by:  irredn0  18749  irredrmul  18753
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