Theorem isirred 20367

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 6876	. . . 4 ⊢ (𝑋 ∈ (Irred‘𝑅) → 𝑅 ∈ dom Irred)
2		irred.3	. . . 4 ⊢ 𝐼 = (Irred‘𝑅)
3	1, 2	eleq2s 2855	. . 3 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ dom Irred)
4	3	elexd 3466	. 2 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ V)
5		eldifi 4085	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) → 𝑋 ∈ 𝐵)
6		irred.4	. . . . . 6 ⊢ 𝑁 = (𝐵 ∖ 𝑈)
7	5, 6	eleq2s 2855	. . . . 5 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵)
8		irred.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
9	7, 8	eleqtrdi 2847	. . . 4 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝑅))
10	9	elfvexd 6878	. . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑅 ∈ V)
11	10	adantr 480	. 2 ⊢ ((𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋) → 𝑅 ∈ V)
12		fvex 6855	. . . . . . . 8 ⊢ (Base‘𝑟) ∈ V
13		difexg 5276	. . . . . . . 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 6846	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = (Base‘𝑅))
18	17, 8	eqtr4di 2790	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = 𝐵)
19	16	fveq2d 6846	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = (Unit‘𝑅))
20		irred.2	. . . . . . . . . . . 12 ⊢ 𝑈 = (Unit‘𝑅)
21	19, 20	eqtr4di 2790	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = 𝑈)
22	18, 21	difeq12d 4081	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = (𝐵 ∖ 𝑈))
23	22, 6	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = 𝑁)
24	15, 23	eqtrd 2772	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = 𝑁)
25	16	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r‘𝑟) = (.r‘𝑅))
26		irred.5	. . . . . . . . . . . . 13 ⊢ · = (.r‘𝑅)
27	25, 26	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r‘𝑟) = · )
28	27	oveqd 7385	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
29	28	neeq1d 2992	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑧))
30	24, 29	raleqbidv 3318	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧))
31	24, 30	raleqbidv 3318	. . . . . . . 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 3419	. . . . . . 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 3887	. . . . . 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 20307	. . . . . 6 ⊢ Irred = (𝑟 ∈ V ↦ ⦋((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧})
35		fvex 6855	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
36	8, 35	eqeltri 2833	. . . . . . . . 9 ⊢ 𝐵 ∈ V
37	36	difexi 5277	. . . . . . . 8 ⊢ (𝐵 ∖ 𝑈) ∈ V
38	6, 37	eqeltri 2833	. . . . . . 7 ⊢ 𝑁 ∈ V
39	38	rabex 5286	. . . . . 6 ⊢ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧} ∈ V
40	33, 34, 39	fvmpt 6949	. . . . 5 ⊢ (𝑅 ∈ V → (Irred‘𝑅) = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
41	2, 40	eqtrid 2784	. . . 4 ⊢ (𝑅 ∈ V → 𝐼 = {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧})
42	41	eleq2d 2823	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧}))
43		neeq2 2996	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝑥 · 𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑋))
44	43	2ralbidv 3202	. . . 4 ⊢ (𝑧 = 𝑋 → (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
45	44	elrab 3648	. . 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3401  Vcvv 3442  ⦋csb 3851   ∖ cdif 3900  dom cdm 5632  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  ._rcmulr 17190  Unitcui 20303  Irredcir 20304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-irred 20307

This theorem is referenced by:  isnirred  20368  isirred2  20369  opprirred  20370  mxidlirredi  33563  rprmirred  33623  ply1dg3rt0irred  33676