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Theorem isirred 20233
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 𝐡 = (Baseβ€˜π‘…)
irred.2 π‘ˆ = (Unitβ€˜π‘…)
irred.3 𝐼 = (Irredβ€˜π‘…)
irred.4 𝑁 = (𝐡 βˆ– π‘ˆ)
irred.5 Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
isirred (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
Distinct variable groups:   π‘₯,𝑦,𝑁   π‘₯,𝑅,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   Β· (π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐼(π‘₯,𝑦)

Proof of Theorem isirred
Dummy variables π‘Ÿ 𝑏 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6929 . . . 4 (𝑋 ∈ (Irredβ€˜π‘…) β†’ 𝑅 ∈ dom Irred)
2 irred.3 . . . 4 𝐼 = (Irredβ€˜π‘…)
31, 2eleq2s 2852 . . 3 (𝑋 ∈ 𝐼 β†’ 𝑅 ∈ dom Irred)
43elexd 3495 . 2 (𝑋 ∈ 𝐼 β†’ 𝑅 ∈ V)
5 eldifi 4127 . . . . . 6 (𝑋 ∈ (𝐡 βˆ– π‘ˆ) β†’ 𝑋 ∈ 𝐡)
6 irred.4 . . . . . 6 𝑁 = (𝐡 βˆ– π‘ˆ)
75, 6eleq2s 2852 . . . . 5 (𝑋 ∈ 𝑁 β†’ 𝑋 ∈ 𝐡)
8 irred.1 . . . . 5 𝐡 = (Baseβ€˜π‘…)
97, 8eleqtrdi 2844 . . . 4 (𝑋 ∈ 𝑁 β†’ 𝑋 ∈ (Baseβ€˜π‘…))
109elfvexd 6931 . . 3 (𝑋 ∈ 𝑁 β†’ 𝑅 ∈ V)
1110adantr 482 . 2 ((𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋) β†’ 𝑅 ∈ V)
12 fvex 6905 . . . . . . . 8 (Baseβ€˜π‘Ÿ) ∈ V
13 difexg 5328 . . . . . . . 8 ((Baseβ€˜π‘Ÿ) ∈ V β†’ ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ)) ∈ V)
1412, 13mp1i 13 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ)) ∈ V)
15 simpr 486 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ)))
16 simpl 484 . . . . . . . . . . . . 13 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ π‘Ÿ = 𝑅)
1716fveq2d 6896 . . . . . . . . . . . 12 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
1817, 8eqtr4di 2791 . . . . . . . . . . 11 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
1916fveq2d 6896 . . . . . . . . . . . 12 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
20 irred.2 . . . . . . . . . . . 12 π‘ˆ = (Unitβ€˜π‘…)
2119, 20eqtr4di 2791 . . . . . . . . . . 11 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
2218, 21difeq12d 4124 . . . . . . . . . 10 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ)) = (𝐡 βˆ– π‘ˆ))
2322, 6eqtr4di 2791 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ)) = 𝑁)
2415, 23eqtrd 2773 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ 𝑏 = 𝑁)
2516fveq2d 6896 . . . . . . . . . . . . 13 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
26 irred.5 . . . . . . . . . . . . 13 Β· = (.rβ€˜π‘…)
2725, 26eqtr4di 2791 . . . . . . . . . . . 12 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (.rβ€˜π‘Ÿ) = Β· )
2827oveqd 7426 . . . . . . . . . . 11 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (π‘₯(.rβ€˜π‘Ÿ)𝑦) = (π‘₯ Β· 𝑦))
2928neeq1d 3001 . . . . . . . . . 10 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ ((π‘₯(.rβ€˜π‘Ÿ)𝑦) β‰  𝑧 ↔ (π‘₯ Β· 𝑦) β‰  𝑧))
3024, 29raleqbidv 3343 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) β‰  𝑧 ↔ βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧))
3124, 30raleqbidv 3343 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) β‰  𝑧 ↔ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧))
3224, 31rabeqbidv 3450 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = ((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ))) β†’ {𝑧 ∈ 𝑏 ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) β‰  𝑧} = {𝑧 ∈ 𝑁 ∣ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧})
3314, 32csbied 3932 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ⦋((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ)) / π‘β¦Œ{𝑧 ∈ 𝑏 ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) β‰  𝑧} = {𝑧 ∈ 𝑁 ∣ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧})
34 df-irred 20173 . . . . . 6 Irred = (π‘Ÿ ∈ V ↦ ⦋((Baseβ€˜π‘Ÿ) βˆ– (Unitβ€˜π‘Ÿ)) / π‘β¦Œ{𝑧 ∈ 𝑏 ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) β‰  𝑧})
35 fvex 6905 . . . . . . . . . 10 (Baseβ€˜π‘…) ∈ V
368, 35eqeltri 2830 . . . . . . . . 9 𝐡 ∈ V
3736difexi 5329 . . . . . . . 8 (𝐡 βˆ– π‘ˆ) ∈ V
386, 37eqeltri 2830 . . . . . . 7 𝑁 ∈ V
3938rabex 5333 . . . . . 6 {𝑧 ∈ 𝑁 ∣ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧} ∈ V
4033, 34, 39fvmpt 6999 . . . . 5 (𝑅 ∈ V β†’ (Irredβ€˜π‘…) = {𝑧 ∈ 𝑁 ∣ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧})
412, 40eqtrid 2785 . . . 4 (𝑅 ∈ V β†’ 𝐼 = {𝑧 ∈ 𝑁 ∣ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧})
4241eleq2d 2820 . . 3 (𝑅 ∈ V β†’ (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ {𝑧 ∈ 𝑁 ∣ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧}))
43 neeq2 3005 . . . . 5 (𝑧 = 𝑋 β†’ ((π‘₯ Β· 𝑦) β‰  𝑧 ↔ (π‘₯ Β· 𝑦) β‰  𝑋))
44432ralbidv 3219 . . . 4 (𝑧 = 𝑋 β†’ (βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧 ↔ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
4544elrab 3684 . . 3 (𝑋 ∈ {𝑧 ∈ 𝑁 ∣ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑧} ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
4642, 45bitrdi 287 . 2 (𝑅 ∈ V β†’ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋)))
474, 11, 46pm5.21nii 380 1 (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433  Vcvv 3475  β¦‹csb 3894   βˆ– cdif 3946  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  Unitcui 20169  Irredcir 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-irred 20173
This theorem is referenced by:  isnirred  20234  isirred2  20235  opprirred  20236  mxidlirredi  32587
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