Theorem isnirred 20380

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 2826	. . . . . 6 ⊢ (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ (𝐵 ∖ 𝑈))
3		eldif 3936	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
4	2, 3	bitri 275	. . . . 5 ⊢ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
5	4	baibr 536	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝑈 ↔ 𝑋 ∈ 𝑁))
6		df-ne 2933	. . . . . . . . 9 ⊢ ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
7	6	ralbii 3082	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑦 ∈ 𝑁 ¬ (𝑥 · 𝑦) = 𝑋)
8		ralnex 3062	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑁 ¬ (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
9	7, 8	bitri 275	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
10	9	ralbii 3082	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥 ∈ 𝑁 ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
11		ralnex 3062	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑁 ¬ ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋 ↔ ¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)
12	10, 11	bitr2i 276	. . . . 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 632	. . 3 ⊢ (𝑋 ∈ 𝐵 → ((¬ 𝑋 ∈ 𝑈 ∧ ¬ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋) ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
15		ioran 985	. . 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 20379	. . 3 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
21	14, 15, 20	3bitr4g 314	. 2 ⊢ (𝑋 ∈ 𝐵 → (¬ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋) ↔ 𝑋 ∈ 𝐼))
22	21	con1bid 355	1 ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∖ cdif 3923  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  ._rcmulr 17272  Unitcui 20315  Irredcir 20316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-irred 20319

This theorem is referenced by:  irredn0  20383  irredrmul  20387