MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isirred Structured version   Visualization version   GIF version

Theorem isirred 18470
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 6114 . . . 4 (𝑋 ∈ (Irred‘𝑅) → 𝑅 ∈ dom Irred)
2 irred.3 . . . 4 𝐼 = (Irred‘𝑅)
31, 2eleq2s 2705 . . 3 (𝑋𝐼𝑅 ∈ dom Irred)
4 elex 3184 . . 3 (𝑅 ∈ dom Irred → 𝑅 ∈ V)
53, 4syl 17 . 2 (𝑋𝐼𝑅 ∈ V)
6 eldifi 3693 . . . . . 6 (𝑋 ∈ (𝐵𝑈) → 𝑋𝐵)
7 irred.4 . . . . . 6 𝑁 = (𝐵𝑈)
86, 7eleq2s 2705 . . . . 5 (𝑋𝑁𝑋𝐵)
9 irred.1 . . . . 5 𝐵 = (Base‘𝑅)
108, 9syl6eleq 2697 . . . 4 (𝑋𝑁𝑋 ∈ (Base‘𝑅))
1110elfvexd 6116 . . 3 (𝑋𝑁𝑅 ∈ V)
1211adantr 479 . 2 ((𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋) → 𝑅 ∈ V)
13 fvex 6097 . . . . . . . 8 (Base‘𝑟) ∈ V
14 difexg 4729 . . . . . . . 8 ((Base‘𝑟) ∈ V → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
1513, 14mp1i 13 . . . . . . 7 (𝑟 = 𝑅 → ((Base‘𝑟) ∖ (Unit‘𝑟)) ∈ V)
16 simpr 475 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟)))
17 simpl 471 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑟 = 𝑅)
1817fveq2d 6091 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = (Base‘𝑅))
1918, 9syl6eqr 2661 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Base‘𝑟) = 𝐵)
2017fveq2d 6091 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = (Unit‘𝑅))
21 irred.2 . . . . . . . . . . . 12 𝑈 = (Unit‘𝑅)
2220, 21syl6eqr 2661 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (Unit‘𝑟) = 𝑈)
2319, 22difeq12d 3690 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = (𝐵𝑈))
2423, 7syl6eqr 2661 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((Base‘𝑟) ∖ (Unit‘𝑟)) = 𝑁)
2516, 24eqtrd 2643 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → 𝑏 = 𝑁)
2617fveq2d 6091 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r𝑟) = (.r𝑅))
27 irred.5 . . . . . . . . . . . . 13 · = (.r𝑅)
2826, 27syl6eqr 2661 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (.r𝑟) = · )
2928oveqd 6543 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
3029neeq1d 2840 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → ((𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑧))
3125, 30raleqbidv 3128 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧))
3225, 31raleqbidv 3128 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → (∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧))
3325, 32rabeqbidv 3167 . . . . . . 7 ((𝑟 = 𝑅𝑏 = ((Base‘𝑟) ∖ (Unit‘𝑟))) → {𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧} = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
3415, 33csbied 3525 . . . . . 6 (𝑟 = 𝑅((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧} = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
35 df-irred 18414 . . . . . 6 Irred = (𝑟 ∈ V ↦ ((Base‘𝑟) ∖ (Unit‘𝑟)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑟)𝑦) ≠ 𝑧})
36 fvex 6097 . . . . . . . . . 10 (Base‘𝑅) ∈ V
379, 36eqeltri 2683 . . . . . . . . 9 𝐵 ∈ V
38 difexg 4729 . . . . . . . . 9 (𝐵 ∈ V → (𝐵𝑈) ∈ V)
3937, 38ax-mp 5 . . . . . . . 8 (𝐵𝑈) ∈ V
407, 39eqeltri 2683 . . . . . . 7 𝑁 ∈ V
4140rabex 4734 . . . . . 6 {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧} ∈ V
4234, 35, 41fvmpt 6175 . . . . 5 (𝑅 ∈ V → (Irred‘𝑅) = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
432, 42syl5eq 2655 . . . 4 (𝑅 ∈ V → 𝐼 = {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧})
4443eleq2d 2672 . . 3 (𝑅 ∈ V → (𝑋𝐼𝑋 ∈ {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧}))
45 neeq2 2844 . . . . 5 (𝑧 = 𝑋 → ((𝑥 · 𝑦) ≠ 𝑧 ↔ (𝑥 · 𝑦) ≠ 𝑋))
46452ralbidv 2971 . . . 4 (𝑧 = 𝑋 → (∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
4746elrab 3330 . . 3 (𝑋 ∈ {𝑧𝑁 ∣ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑧} ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
4844, 47syl6bb 274 . 2 (𝑅 ∈ V → (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋)))
495, 12, 48pm5.21nii 366 1 (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  {crab 2899  Vcvv 3172  csb 3498  cdif 3536  dom cdm 5027  cfv 5789  (class class class)co 6526  Basecbs 15643  .rcmulr 15717  Unitcui 18410  Irredcir 18411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-ov 6529  df-irred 18414
This theorem is referenced by:  isnirred  18471  isirred2  18472  opprirred  18473
  Copyright terms: Public domain W3C validator