Theorem isnirred 20535

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 2858	. . . . . 6 ⊢ (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ (𝐵 ∖ 𝑈))
3		eldif 3918	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
4	2, 3	bitri 278	. . . . 5 ⊢ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
5	4	baibr 546	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝑈 ↔ 𝑋 ∈ 𝑁))
6		df-ne 2962	. . . . . . . . 9 ⊢ ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
7	6	ralbii 3114	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦 ∈ 𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8		ralnex 3094	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
9	7, 8	bitri 278	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
10	9	ralbii 3114	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥 ∈ 𝑁 ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
11		ralnex 3094	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑁 ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
12	10, 11	bitr2i 279	. . . . 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 644	. . 3 ⊢ (𝑋 ∈ 𝐵 → ((¬ 𝑋 ∈ 𝑈 ∧ ¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15		ioran 999	. . 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 20534	. . 3 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
21	14, 15, 20	3bitr4g 317	. 2 ⊢ (𝑋 ∈ 𝐵 → (¬ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋) ↔ 𝑋 ∈ 𝐼))
22	21	con1bid 358	1 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092   ∖ cdif 3905  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  ._rcmulr 17336  Unitcui 20470  Irredcir 20471

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-irred 20474

This theorem is used by:  irredn0  20538  irredrmul  20542