Theorem isirred 20379

Description: An irreducible element of a ring is a non-unit that is not the product of two non-units. (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
isirred	⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀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 isirred

Dummy variables 𝑟 𝑏 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6913	. . . 4 ⊢ (𝑋 ∈ (Irred‘𝑅) → 𝑅 ∈ dom Irred)
2		irred.3	. . . 4 ⊢ 𝐼 = (Irred‘𝑅)
3	1, 2	eleq2s 2852	. . 3 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ dom Irred)
4	3	elexd 3483	. 2 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ V)
5		eldifi 4106	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) → 𝑋 ∈ 𝐵)
6		irred.4	. . . . . 6 ⊢ 𝑁 = (𝐵 ∖ 𝑈)
7	5, 6	eleq2s 2852	. . . . 5 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵)
8		irred.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
9	7, 8	eleqtrdi 2844	. . . 4 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝑅))
10	9	elfvexd 6915	. . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑅 ∈ V)
11	10	adantr 480	. 2 ⊢ ((𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋) → 𝑅 ∈ V)
12		fvex 6889	. . . . . . . 8 ⊢ (Base‘𝑟) ∈ V
13		difexg 5299	. . . . . . . 8 ⊢ ((Base‘𝑟) ∈ V → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
14	12, 13	mp1i 13	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
15		simpr 484	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟)))
16		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑟 = 𝑅)
17	16	fveq2d 6880	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = (Base‘𝑅))
18	17, 8	eqtr4di 2788	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = 𝐵)
19	16	fveq2d 6880	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = (Unit‘𝑅))
20		irred.2	. . . . . . . . . . . 12 ⊢ 𝑈 = (Unit‘𝑅)
21	19, 20	eqtr4di 2788	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = 𝑈)
22	18, 21	difeq12d 4102	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = (𝐵 ∖ 𝑈))
23	22, 6	eqtr4di 2788	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = 𝑁)
24	15, 23	eqtrd 2770	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = 𝑁)
25	16	fveq2d 6880	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r‘𝑟) = (.r‘𝑅))
26		irred.5	. . . . . . . . . . . . 13 ⊢ · = (.r‘𝑅)
27	25, 26	eqtr4di 2788	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r‘𝑟) = · )
28	27	oveqd 7422	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
29	28	neeq1d 2991	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑧))
30	24, 29	raleqbidv 3325	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧))
31	24, 30	raleqbidv 3325	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧))
32	24, 31	rabeqbidv 3434	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → {𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧} = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
33	14, 32	csbied 3910	. . . . . 6 ⊢ (𝑟 = 𝑅 → ⦋((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧} = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
34		df-irred 20319	. . . . . 6 ⊢ Irred = (𝑟 ∈ V ↦ ⦋((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧})
35		fvex 6889	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
36	8, 35	eqeltri 2830	. . . . . . . . 9 ⊢ 𝐵 ∈ V
37	36	difexi 5300	. . . . . . . 8 ⊢ (𝐵 ∖ 𝑈) ∈ V
38	6, 37	eqeltri 2830	. . . . . . 7 ⊢ 𝑁 ∈ V
39	38	rabex 5309	. . . . . 6 ⊢ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧} ∈ V
40	33, 34, 39	fvmpt 6986	. . . . 5 ⊢ (𝑅 ∈ V → (Irred‘𝑅) = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
41	2, 40	eqtrid 2782	. . . 4 ⊢ (𝑅 ∈ V → 𝐼 = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
42	41	eleq2d 2820	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧}))
43		neeq2 2995	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝑥 · 𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑋))
44	43	2ralbidv 3205	. . . 4 ⊢ (𝑧 = 𝑋 → (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
45	44	elrab 3671	. . 3 ⊢ (𝑋 ∈ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧} ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
46	42, 45	bitrdi 287	. 2 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
47	4, 11, 46	pm5.21nii 378	1 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  {crab 3415  Vcvv 3459  ⦋csb 3874   ∖ cdif 3923  dom cdm 5654  ‘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:  isnirred  20380  isirred2  20381  opprirred  20382  mxidlirredi  33486  rprmirred  33546  ply1dg3rt0irred  33595