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Theorem isirred 15481
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  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
isirred  |-  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) )
Distinct variable groups:    x, y, N    x, R, y    x, X, y
Allowed substitution hints:    B( x, y)    .x. ( 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 5554 . . . 4  |-  ( X  e.  (Irred `  R
)  ->  R  e.  dom Irred )
2 irred.3 . . . 4  |-  I  =  (Irred `  R )
31, 2eleq2s 2375 . . 3  |-  ( X  e.  I  ->  R  e.  dom Irred )
4 elex 2796 . . 3  |-  ( R  e.  dom Irred  ->  R  e. 
_V )
53, 4syl 15 . 2  |-  ( X  e.  I  ->  R  e.  _V )
6 eldifi 3298 . . . . . 6  |-  ( X  e.  ( B  \  U )  ->  X  e.  B )
7 irred.4 . . . . . 6  |-  N  =  ( B  \  U
)
86, 7eleq2s 2375 . . . . 5  |-  ( X  e.  N  ->  X  e.  B )
9 irred.1 . . . . 5  |-  B  =  ( Base `  R
)
108, 9syl6eleq 2373 . . . 4  |-  ( X  e.  N  ->  X  e.  ( Base `  R
) )
11 elfvdm 5554 . . . 4  |-  ( X  e.  ( Base `  R
)  ->  R  e.  dom  Base )
12 elex 2796 . . . 4  |-  ( R  e.  dom  Base  ->  R  e.  _V )
1310, 11, 123syl 18 . . 3  |-  ( X  e.  N  ->  R  e.  _V )
1413adantr 451 . 2  |-  ( ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  X )  ->  R  e.  _V )
15 fvex 5539 . . . . . . . 8  |-  ( Base `  r )  e.  _V
16 difexg 4162 . . . . . . . 8  |-  ( (
Base `  r )  e.  _V  ->  ( ( Base `  r )  \ 
(Unit `  r )
)  e.  _V )
1715, 16mp1i 11 . . . . . . 7  |-  ( r  =  R  ->  (
( Base `  r )  \  (Unit `  r )
)  e.  _V )
18 simpr 447 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  b  =  ( ( Base `  r )  \  (Unit `  r ) ) )
19 simpl 443 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  r  =  R )
2019fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  ( Base `  R
) )
2120, 9syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  B )
2219fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  (Unit `  R ) )
23 irred.2 . . . . . . . . . . . 12  |-  U  =  (Unit `  R )
2422, 23syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  U )
2521, 24difeq12d 3295 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  ( B 
\  U ) )
2625, 7syl6eqr 2333 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  N )
2718, 26eqtrd 2315 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  b  =  N )
2819fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  =  ( .r `  R
) )
29 irred.5 . . . . . . . . . . . . 13  |-  .x.  =  ( .r `  R )
3028, 29syl6eqr 2333 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  = 
.x.  )
3130oveqd 5875 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
3231neeq1d 2459 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( x ( .r
`  r ) y )  =/=  z  <->  ( x  .x.  y )  =/=  z
) )
3327, 32raleqbidv 2748 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( A. y  e.  b 
( x ( .r
`  r ) y )  =/=  z  <->  A. y  e.  N  ( x  .x.  y )  =/=  z
) )
3427, 33raleqbidv 2748 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( A. x  e.  b  A. y  e.  b 
( x ( .r
`  r ) y )  =/=  z  <->  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z
) )
3527, 34rabeqbidv 2783 . . . . . . 7  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  { z  e.  b  |  A. x  e.  b  A. y  e.  b  (
x ( .r `  r ) y )  =/=  z }  =  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
3617, 35csbied 3123 . . . . . 6  |-  ( r  =  R  ->  [_ (
( Base `  r )  \  (Unit `  r )
)  /  b ]_ { z  e.  b  |  A. x  e.  b  A. y  e.  b  ( x ( .r `  r ) y )  =/=  z }  =  { z  e.  N  |  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  z } )
37 df-irred 15425 . . . . . 6  |- Irred  =  ( r  e.  _V  |->  [_ ( ( Base `  r
)  \  (Unit `  r
) )  /  b ]_ { z  e.  b  |  A. x  e.  b  A. y  e.  b  ( x ( .r `  r ) y )  =/=  z } )
38 fvex 5539 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
399, 38eqeltri 2353 . . . . . . . . 9  |-  B  e. 
_V
40 difexg 4162 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  U )  e. 
_V )
4139, 40ax-mp 8 . . . . . . . 8  |-  ( B 
\  U )  e. 
_V
427, 41eqeltri 2353 . . . . . . 7  |-  N  e. 
_V
4342rabex 4165 . . . . . 6  |-  { z  e.  N  |  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  z }  e.  _V
4436, 37, 43fvmpt 5602 . . . . 5  |-  ( R  e.  _V  ->  (Irred `  R )  =  {
z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
452, 44syl5eq 2327 . . . 4  |-  ( R  e.  _V  ->  I  =  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
4645eleq2d 2350 . . 3  |-  ( R  e.  _V  ->  ( X  e.  I  <->  X  e.  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } ) )
47 neeq2 2455 . . . . 5  |-  ( z  =  X  ->  (
( x  .x.  y
)  =/=  z  <->  ( x  .x.  y )  =/=  X
) )
48472ralbidv 2585 . . . 4  |-  ( z  =  X  ->  ( A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z  <->  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) )
4948elrab 2923 . . 3  |-  ( X  e.  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } 
<->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X ) )
5046, 49syl6bb 252 . 2  |-  ( R  e.  _V  ->  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) ) )
515, 14, 50pm5.21nii 342 1  |-  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788   [_csb 3081    \ cdif 3149   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  Unitcui 15421  Irredcir 15422
This theorem is referenced by:  isnirred  15482  isirred2  15483  opprirred  15484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-irred 15425
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