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Theorem isnirred 20125
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 2829 . . . . . 6 (𝑋𝑁𝑋 ∈ (𝐵𝑈))
3 eldif 3919 . . . . . 6 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
42, 3bitri 274 . . . . 5 (𝑋𝑁 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
54baibr 537 . . . 4 (𝑋𝐵 → (¬ 𝑋𝑈𝑋𝑁))
6 df-ne 2943 . . . . . . . . 9 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
76ralbii 3095 . . . . . . . 8 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8 ralnex 3074 . . . . . . . 8 (∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
97, 8bitri 274 . . . . . . 7 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
109ralbii 3095 . . . . . 6 (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
11 ralnex 3074 . . . . . 6 (∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
1210, 11bitr2i 275 . . . . 5 (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)
1312a1i 11 . . . 4 (𝑋𝐵 → (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
145, 13anbi12d 631 . . 3 (𝑋𝐵 → ((¬ 𝑋𝑈 ∧ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15 ioran 982 . . 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 20124 . . 3 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
2114, 15, 203bitr4g 313 . 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 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2942  wral 3063  wrex 3072  cdif 3906  cfv 6494  (class class class)co 7354  Basecbs 17080  .rcmulr 17131  Unitcui 20064  Irredcir 20065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6446  df-fun 6496  df-fv 6502  df-ov 7357  df-irred 20068
This theorem is referenced by:  irredn0  20128  irredrmul  20132
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