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 = (𝐵 ∖ 𝑈)
irred.5	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
isnirred	⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (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

Step	Hyp	Ref	Expression
1		irred.4	. . . . . . 7 ⊢ 𝑁 = (𝐵 ∖ 𝑈)
2	1	eleq2i 2722	. . . . . 6 ⊢ (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ (𝐵 ∖ 𝑈))
3		eldif 3617	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
4	2, 3	bitri 264	. . . . 5 ⊢ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
5	4	baibr 965	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝑈 ↔ 𝑋 ∈ 𝑁))
6		df-ne 2824	. . . . . . . . 9 ⊢ ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
7	6	ralbii 3009	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦 ∈ 𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8		ralnex 3021	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
9	7, 8	bitri 264	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
10	9	ralbii 3009	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥 ∈ 𝑁 ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
11		ralnex 3021	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑁 ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
12	10, 11	bitr2i 265	. . . . 5 ⊢ (¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)
13	12	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
14	5, 13	anbi12d 747	. . 3 ⊢ (𝑋 ∈ 𝐵 → ((¬ 𝑋 ∈ 𝑈 ∧ ¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15		ioran 510	. . 3 ⊢ (¬ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (¬ 𝑋 ∈ 𝑈 ∧ ¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (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 ⊢ · = (.r‘𝑅)
20	16, 17, 18, 1, 19	isirred 18745	. . 3 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
21	14, 15, 20	3bitr4g 303	. 2 ⊢ (𝑋 ∈ 𝐵 → (¬ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋) ↔ 𝑋 ∈ 𝐼))
22	21	con1bid 344	1 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)))

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