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Theorem isirred 18745
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 = (𝐵𝑈)
irred.5 · = (.r𝑅)
Assertion
Ref Expression
isirred (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (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.
StepHypRef Expression
1 elfvdm 6258 . . . 4 (𝑋 ∈ (Irred‘𝑅) → 𝑅 ∈ dom Irred)
2 irred.3 . . . 4 𝐼 = (Irred‘𝑅)
31, 2eleq2s 2748 . . 3 (𝑋𝐼𝑅 ∈ dom Irred)
4 elex 3243 . . 3 (𝑅 ∈ dom Irred → 𝑅 ∈ V)
53, 4syl 17 . 2 (𝑋𝐼𝑅 ∈ V)
6 eldifi 3765 . . . . . 6 (𝑋 ∈ (𝐵𝑈) → 𝑋𝐵)
7 irred.4 . . . . . 6 𝑁 = (𝐵𝑈)
86, 7eleq2s 2748 . . . . 5 (𝑋𝑁𝑋𝐵)
9 irred.1 . . . . 5 𝐵 = (Base‘𝑅)
108, 9syl6eleq 2740 . . . 4 (𝑋𝑁𝑋 ∈ (Base‘𝑅))
1110elfvexd 6260 . . 3 (𝑋𝑁𝑅 ∈ V)
1211adantr 480 . 2 ((𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋) → 𝑅 ∈ V)
13 fvex 6239 . . . . . . . 8 (Base‘𝑟) ∈ V
14 difexg 4841 . . . . . . . 8 ((Base‘𝑟) ∈ V → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
1513, 14mp1i 13 . . . . . . 7 (𝑟 = 𝑅 → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
16 simpr 476 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟)))
17 simpl 472 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑟 = 𝑅)
1817fveq2d 6233 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = (Base‘𝑅))
1918, 9syl6eqr 2703 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = 𝐵)
2017fveq2d 6233 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = (Unit‘𝑅))
21 irred.2 . . . . . . . . . . . 12 𝑈 = (Unit‘𝑅)
2220, 21syl6eqr 2703 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = 𝑈)
2319, 22difeq12d 3762 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = (𝐵𝑈))
2423, 7syl6eqr 2703 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = 𝑁)
2516, 24eqtrd 2685 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = 𝑁)
2617fveq2d 6233 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r𝑟) = (.r𝑅))
27 irred.5 . . . . . . . . . . . . 13 · = (.r𝑅)
2826, 27syl6eqr 2703 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r𝑟) = · )
2928oveqd 6707 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
3029neeq1d 2882 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑧))
3125, 30raleqbidv 3182 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧))
3225, 31raleqbidv 3182 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧))
3325, 32rabeqbidv 3226 . . . . . . 7 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → {𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧} = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
3415, 33csbied 3593 . . . . . 6 (𝑟 = 𝑅((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧} = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
35 df-irred 18689 . . . . . 6 Irred = (𝑟 ∈ V ↦ ((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧})
36 fvex 6239 . . . . . . . . . 10 (Base‘𝑅) ∈ V
379, 36eqeltri 2726 . . . . . . . . 9 𝐵 ∈ V
38 difexg 4841 . . . . . . . . 9 (𝐵 ∈ V → (𝐵𝑈) ∈ V)
3937, 38ax-mp 5 . . . . . . . 8 (𝐵𝑈) ∈ V
407, 39eqeltri 2726 . . . . . . 7 𝑁 ∈ V
4140rabex 4845 . . . . . 6 {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧} ∈ V
4234, 35, 41fvmpt 6321 . . . . 5 (𝑅 ∈ V → (Irred‘𝑅) = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
432, 42syl5eq 2697 . . . 4 (𝑅 ∈ V → 𝐼 = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
4443eleq2d 2716 . . 3 (𝑅 ∈ V → (𝑋𝐼𝑋 ∈ {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧}))
45 neeq2 2886 . . . . 5 (𝑧 = 𝑋 → ((𝑥 · 𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑋))
46452ralbidv 3018 . . . 4 (𝑧 = 𝑋 → (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
4746elrab 3396 . . 3 (𝑋 ∈ {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧} ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
4844, 47syl6bb 276 . 2 (𝑅 ∈ V → (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
495, 12, 48pm5.21nii 367 1 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  Vcvv 3231  csb 3566  cdif 3604  dom cdm 5143  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989  Unitcui 18685  Irredcir 18686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-irred 18689
This theorem is referenced by:  isnirred  18746  isirred2  18747  opprirred  18748
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