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Theorem isirred 20436
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 6944 . . . 4 (𝑋 ∈ (Irred‘𝑅) → 𝑅 ∈ dom Irred)
2 irred.3 . . . 4 𝐼 = (Irred‘𝑅)
31, 2eleq2s 2857 . . 3 (𝑋𝐼𝑅 ∈ dom Irred)
43elexd 3502 . 2 (𝑋𝐼𝑅 ∈ V)
5 eldifi 4141 . . . . . 6 (𝑋 ∈ (𝐵𝑈) → 𝑋𝐵)
6 irred.4 . . . . . 6 𝑁 = (𝐵𝑈)
75, 6eleq2s 2857 . . . . 5 (𝑋𝑁𝑋𝐵)
8 irred.1 . . . . 5 𝐵 = (Base‘𝑅)
97, 8eleqtrdi 2849 . . . 4 (𝑋𝑁𝑋 ∈ (Base‘𝑅))
109elfvexd 6946 . . 3 (𝑋𝑁𝑅 ∈ V)
1110adantr 480 . 2 ((𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋) → 𝑅 ∈ V)
12 fvex 6920 . . . . . . . 8 (Base‘𝑟) ∈ V
13 difexg 5335 . . . . . . . 8 ((Base‘𝑟) ∈ V → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
1412, 13mp1i 13 . . . . . . 7 (𝑟 = 𝑅 → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
15 simpr 484 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟)))
16 simpl 482 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑟 = 𝑅)
1716fveq2d 6911 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = (Base‘𝑅))
1817, 8eqtr4di 2793 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = 𝐵)
1916fveq2d 6911 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = (Unit‘𝑅))
20 irred.2 . . . . . . . . . . . 12 𝑈 = (Unit‘𝑅)
2119, 20eqtr4di 2793 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = 𝑈)
2218, 21difeq12d 4137 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = (𝐵𝑈))
2322, 6eqtr4di 2793 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = 𝑁)
2415, 23eqtrd 2775 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = 𝑁)
2516fveq2d 6911 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r𝑟) = (.r𝑅))
26 irred.5 . . . . . . . . . . . . 13 · = (.r𝑅)
2725, 26eqtr4di 2793 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r𝑟) = · )
2827oveqd 7448 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
2928neeq1d 2998 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑧))
3024, 29raleqbidv 3344 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧))
3124, 30raleqbidv 3344 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧))
3224, 31rabeqbidv 3452 . . . . . . 7 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → {𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧} = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
3314, 32csbied 3946 . . . . . 6 (𝑟 = 𝑅((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧} = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
34 df-irred 20376 . . . . . 6 Irred = (𝑟 ∈ V ↦ ((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧})
35 fvex 6920 . . . . . . . . . 10 (Base‘𝑅) ∈ V
368, 35eqeltri 2835 . . . . . . . . 9 𝐵 ∈ V
3736difexi 5336 . . . . . . . 8 (𝐵𝑈) ∈ V
386, 37eqeltri 2835 . . . . . . 7 𝑁 ∈ V
3938rabex 5345 . . . . . 6 {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧} ∈ V
4033, 34, 39fvmpt 7016 . . . . 5 (𝑅 ∈ V → (Irred‘𝑅) = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
412, 40eqtrid 2787 . . . 4 (𝑅 ∈ V → 𝐼 = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
4241eleq2d 2825 . . 3 (𝑅 ∈ V → (𝑋𝐼𝑋 ∈ {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧}))
43 neeq2 3002 . . . . 5 (𝑧 = 𝑋 → ((𝑥 · 𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑋))
44432ralbidv 3219 . . . 4 (𝑧 = 𝑋 → (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
4544elrab 3695 . . 3 (𝑋 ∈ {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧} ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
4642, 45bitrdi 287 . 2 (𝑅 ∈ V → (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
474, 11, 46pm5.21nii 378 1 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  {crab 3433  Vcvv 3478  csb 3908  cdif 3960  dom cdm 5689  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  Unitcui 20372  Irredcir 20373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-irred 20376
This theorem is referenced by:  isnirred  20437  isirred2  20438  opprirred  20439  mxidlirredi  33479  rprmirred  33539  ply1dg3rt0irred  33587
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