Theorem isirred 18522

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 6130	. . . 4 ⊢ (𝑋 ∈ (Irred‘𝑅) → 𝑅 ∈ dom Irred)
2		irred.3	. . . 4 ⊢ 𝐼 = (Irred‘𝑅)
3	1, 2	eleq2s 2706	. . 3 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ dom Irred)
4		elex 3185	. . 3 ⊢ (𝑅 ∈ dom Irred → 𝑅 ∈ V)
5	3, 4	syl 17	. 2 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ V)
6		eldifi 3694	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) → 𝑋 ∈ 𝐵)
7		irred.4	. . . . . 6 ⊢ 𝑁 = (𝐵 ∖ 𝑈)
8	6, 7	eleq2s 2706	. . . . 5 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵)
9		irred.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
10	8, 9	syl6eleq 2698	. . . 4 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝑅))
11	10	elfvexd 6132	. . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑅 ∈ V)
12	11	adantr 480	. 2 ⊢ ((𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋) → 𝑅 ∈ V)
13		fvex 6113	. . . . . . . 8 ⊢ (Base‘𝑟) ∈ V
14		difexg 4735	. . . . . . . 8 ⊢ ((Base‘𝑟) ∈ V → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
15	13, 14	mp1i 13	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
16		simpr 476	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟)))
17		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑟 = 𝑅)
18	17	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = (Base‘𝑅))
19	18, 9	syl6eqr 2662	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = 𝐵)
20	17	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = (Unit‘𝑅))
21		irred.2	. . . . . . . . . . . 12 ⊢ 𝑈 = (Unit‘𝑅)
22	20, 21	syl6eqr 2662	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = 𝑈)
23	19, 22	difeq12d 3691	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = (𝐵 ∖ 𝑈))
24	23, 7	syl6eqr 2662	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = 𝑁)
25	16, 24	eqtrd 2644	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = 𝑁)
26	17	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r‘𝑟) = (.r‘𝑅))
27		irred.5	. . . . . . . . . . . . 13 ⊢ · = (.r‘𝑅)
28	26, 27	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r‘𝑟) = · )
29	28	oveqd 6566	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
30	29	neeq1d 2841	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑧))
31	25, 30	raleqbidv 3129	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧))
32	25, 31	raleqbidv 3129	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧))
33	25, 32	rabeqbidv 3168	. . . . . . 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})
34	15, 33	csbied 3526	. . . . . 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})
35		df-irred 18466	. . . . . 6 ⊢ Irred = (𝑟 ∈ V ↦ ⦋((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧})
36		fvex 6113	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
37	9, 36	eqeltri 2684	. . . . . . . . 9 ⊢ 𝐵 ∈ V
38		difexg 4735	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∖ 𝑈) ∈ V)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (𝐵 ∖ 𝑈) ∈ V
40	7, 39	eqeltri 2684	. . . . . . 7 ⊢ 𝑁 ∈ V
41	40	rabex 4740	. . . . . 6 ⊢ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧} ∈ V
42	34, 35, 41	fvmpt 6191	. . . . 5 ⊢ (𝑅 ∈ V → (Irred‘𝑅) = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
43	2, 42	syl5eq 2656	. . . 4 ⊢ (𝑅 ∈ V → 𝐼 = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
44	43	eleq2d 2673	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧}))
45		neeq2 2845	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝑥 · 𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑋))
46	45	2ralbidv 2972	. . . 4 ⊢ (𝑧 = 𝑋 → (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
47	46	elrab 3331	. . 3 ⊢ (𝑋 ∈ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧} ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
48	44, 47	syl6bb 275	. 2 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
49	5, 12, 48	pm5.21nii 367	1 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ⦋csb 3499   ∖ cdif 3537  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  ._rcmulr 15769  Unitcui 18462  Irredcir 18463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-irred 18466

This theorem is referenced by:  isnirred  18523  isirred2  18524  opprirred  18525