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Theorem isnirred 19185
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 2850 . . . . . 6 (𝑋𝑁𝑋 ∈ (𝐵𝑈))
3 eldif 3832 . . . . . 6 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
42, 3bitri 267 . . . . 5 (𝑋𝑁 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
54baibr 529 . . . 4 (𝑋𝐵 → (¬ 𝑋𝑈𝑋𝑁))
6 df-ne 2961 . . . . . . . . 9 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
76ralbii 3108 . . . . . . . 8 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8 ralnex 3176 . . . . . . . 8 (∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
97, 8bitri 267 . . . . . . 7 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
109ralbii 3108 . . . . . 6 (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
11 ralnex 3176 . . . . . 6 (∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
1210, 11bitr2i 268 . . . . 5 (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)
1312a1i 11 . . . 4 (𝑋𝐵 → (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
145, 13anbi12d 622 . . 3 (𝑋𝐵 → ((¬ 𝑋𝑈 ∧ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15 ioran 967 . . 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 19184 . . 3 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
2114, 15, 203bitr4g 306 . 2 (𝑋𝐵 → (¬ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ 𝑋𝐼))
2221con1bid 348 1 (𝑋𝐵 → (¬ 𝑋𝐼 ↔ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 834   = wceq 1508  wcel 2051  wne 2960  wral 3081  wrex 3082  cdif 3819  cfv 6185  (class class class)co 6974  Basecbs 16337  .rcmulr 16420  Unitcui 19124  Irredcir 19125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-ov 6977  df-irred 19128
This theorem is referenced by:  irredn0  19188  irredrmul  19192
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