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Theorem isnirred 19450
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 2884 . . . . . 6 (𝑋𝑁𝑋 ∈ (𝐵𝑈))
3 eldif 3894 . . . . . 6 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
42, 3bitri 278 . . . . 5 (𝑋𝑁 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
54baibr 540 . . . 4 (𝑋𝐵 → (¬ 𝑋𝑈𝑋𝑁))
6 df-ne 2991 . . . . . . . . 9 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
76ralbii 3136 . . . . . . . 8 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8 ralnex 3202 . . . . . . . 8 (∀𝑦𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
97, 8bitri 278 . . . . . . 7 (∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
109ralbii 3136 . . . . . 6 (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
11 ralnex 3202 . . . . . 6 (∀𝑥𝑁 ¬ ∃𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)
1210, 11bitr2i 279 . . . . 5 (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)
1312a1i 11 . . . 4 (𝑋𝐵 → (¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
145, 13anbi12d 633 . . 3 (𝑋𝐵 → ((¬ 𝑋𝑈 ∧ ¬ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15 ioran 981 . . 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 19449 . . 3 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
2114, 15, 203bitr4g 317 . 2 (𝑋𝐵 → (¬ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋) ↔ 𝑋𝐼))
2221con1bid 359 1 (𝑋𝐵 → (¬ 𝑋𝐼 ↔ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  cdif 3881  cfv 6328  (class class class)co 7139  Basecbs 16479  .rcmulr 16562  Unitcui 19389  Irredcir 19390
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-irred 19393
This theorem is referenced by:  irredn0  19453  irredrmul  19457
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