Theorem isirred 20225

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 6925	. . . 4 ⊢ (𝑋 ∈ (Irred‘𝑅) → 𝑅 ∈ dom Irred)
2		irred.3	. . . 4 ⊢ 𝐼 = (Irred‘𝑅)
3	1, 2	eleq2s 2851	. . 3 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ dom Irred)
4	3	elexd 3494	. 2 ⊢ (𝑋 ∈ 𝐼 → 𝑅 ∈ V)
5		eldifi 4125	. . . . . 6 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) → 𝑋 ∈ 𝐵)
6		irred.4	. . . . . 6 ⊢ 𝑁 = (𝐵 ∖ 𝑈)
7	5, 6	eleq2s 2851	. . . . 5 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵)
8		irred.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
9	7, 8	eleqtrdi 2843	. . . 4 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝑅))
10	9	elfvexd 6927	. . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑅 ∈ V)
11	10	adantr 481	. 2 ⊢ ((𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋) → 𝑅 ∈ V)
12		fvex 6901	. . . . . . . 8 ⊢ (Base‘𝑟) ∈ V
13		difexg 5326	. . . . . . . 8 ⊢ ((Base‘𝑟) ∈ V → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
14	12, 13	mp1i 13	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
15		simpr 485	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟)))
16		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑟 = 𝑅)
17	16	fveq2d 6892	. . . . . . . . . . . 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 6892	. . . . . . . . . . . 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 4122	. . . . . . . . . 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 6892	. . . . . . . . . . . . 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 7422	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
29	28	neeq1d 3000	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑧))
30	24, 29	raleqbidv 3342	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧))
31	24, 30	raleqbidv 3342	. . . . . . . 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 3449	. . . . . . 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 3930	. . . . . 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 20165	. . . . . 6 ⊢ Irred = (𝑟 ∈ V ↦ ⦋((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ≠ 𝑧})
35		fvex 6901	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
36	8, 35	eqeltri 2829	. . . . . . . . 9 ⊢ 𝐵 ∈ V
37	36	difexi 5327	. . . . . . . 8 ⊢ (𝐵 ∖ 𝑈) ∈ V
38	6, 37	eqeltri 2829	. . . . . . 7 ⊢ 𝑁 ∈ V
39	38	rabex 5331	. . . . . 6 ⊢ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧} ∈ V
40	33, 34, 39	fvmpt 6995	. . . . 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 2819	. . 3 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧}))
43		neeq2 3004	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝑥 · 𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑋))
44	43	2ralbidv 3218	. . . 4 ⊢ (𝑧 = 𝑋 → (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧 ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
45	44	elrab 3682	. . 3 ⊢ (𝑋 ∈ {𝑧 ∈ 𝑁 ∣ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑧} ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))
46	42, 45	bitrdi 286	. 2 ⊢ (𝑅 ∈ V → (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
47	4, 11, 46	pm5.21nii 379	1 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  {crab 3432  Vcvv 3474  ⦋csb 3892   ∖ cdif 3944  dom cdm 5675  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  ._rcmulr 17194  Unitcui 20161  Irredcir 20162

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-irred 20165

This theorem is referenced by:  isnirred  20226  isirred2  20227  opprirred  20228  mxidlirredi  32575