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Theorem isirred 17149
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 5892 . . . 4  |-  ( X  e.  (Irred `  R
)  ->  R  e.  dom Irred )
2 irred.3 . . . 4  |-  I  =  (Irred `  R )
31, 2eleq2s 2575 . . 3  |-  ( X  e.  I  ->  R  e.  dom Irred )
4 elex 3122 . . 3  |-  ( R  e.  dom Irred  ->  R  e. 
_V )
53, 4syl 16 . 2  |-  ( X  e.  I  ->  R  e.  _V )
6 eldifi 3626 . . . . . 6  |-  ( X  e.  ( B  \  U )  ->  X  e.  B )
7 irred.4 . . . . . 6  |-  N  =  ( B  \  U
)
86, 7eleq2s 2575 . . . . 5  |-  ( X  e.  N  ->  X  e.  B )
9 irred.1 . . . . 5  |-  B  =  ( Base `  R
)
108, 9syl6eleq 2565 . . . 4  |-  ( X  e.  N  ->  X  e.  ( Base `  R
) )
1110elfvexd 5894 . . 3  |-  ( X  e.  N  ->  R  e.  _V )
1211adantr 465 . 2  |-  ( ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  X )  ->  R  e.  _V )
13 fvex 5876 . . . . . . . 8  |-  ( Base `  r )  e.  _V
14 difexg 4595 . . . . . . . 8  |-  ( (
Base `  r )  e.  _V  ->  ( ( Base `  r )  \ 
(Unit `  r )
)  e.  _V )
1513, 14mp1i 12 . . . . . . 7  |-  ( r  =  R  ->  (
( Base `  r )  \  (Unit `  r )
)  e.  _V )
16 simpr 461 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  b  =  ( ( Base `  r )  \  (Unit `  r ) ) )
17 simpl 457 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  r  =  R )
1817fveq2d 5870 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  ( Base `  R
) )
1918, 9syl6eqr 2526 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  B )
2017fveq2d 5870 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  (Unit `  R ) )
21 irred.2 . . . . . . . . . . . 12  |-  U  =  (Unit `  R )
2220, 21syl6eqr 2526 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  U )
2319, 22difeq12d 3623 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  ( B 
\  U ) )
2423, 7syl6eqr 2526 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  N )
2516, 24eqtrd 2508 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  b  =  N )
2617fveq2d 5870 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  =  ( .r `  R
) )
27 irred.5 . . . . . . . . . . . . 13  |-  .x.  =  ( .r `  R )
2826, 27syl6eqr 2526 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  = 
.x.  )
2928oveqd 6301 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
3029neeq1d 2744 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( x ( .r
`  r ) y )  =/=  z  <->  ( x  .x.  y )  =/=  z
) )
3125, 30raleqbidv 3072 . . . . . . . . 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
) )
3225, 31raleqbidv 3072 . . . . . . . 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
) )
3325, 32rabeqbidv 3108 . . . . . . 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 } )
3415, 33csbied 3462 . . . . . 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 } )
35 df-irred 17093 . . . . . 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 } )
36 fvex 5876 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
379, 36eqeltri 2551 . . . . . . . . 9  |-  B  e. 
_V
38 difexg 4595 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  U )  e. 
_V )
3937, 38ax-mp 5 . . . . . . . 8  |-  ( B 
\  U )  e. 
_V
407, 39eqeltri 2551 . . . . . . 7  |-  N  e. 
_V
4140rabex 4598 . . . . . 6  |-  { z  e.  N  |  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  z }  e.  _V
4234, 35, 41fvmpt 5950 . . . . 5  |-  ( R  e.  _V  ->  (Irred `  R )  =  {
z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
432, 42syl5eq 2520 . . . 4  |-  ( R  e.  _V  ->  I  =  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
4443eleq2d 2537 . . 3  |-  ( R  e.  _V  ->  ( X  e.  I  <->  X  e.  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } ) )
45 neeq2 2750 . . . . 5  |-  ( z  =  X  ->  (
( x  .x.  y
)  =/=  z  <->  ( x  .x.  y )  =/=  X
) )
46452ralbidv 2908 . . . 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
) )
4746elrab 3261 . . 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 ) )
4844, 47syl6bb 261 . 2  |-  ( R  e.  _V  ->  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) ) )
495, 12, 48pm5.21nii 353 1  |-  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  X
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   [_csb 3435    \ cdif 3473   dom cdm 4999   ` cfv 5588  (class class class)co 6284   Basecbs 14490   .rcmulr 14556  Unitcui 17089  Irredcir 17090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-irred 17093
This theorem is referenced by:  isnirred  17150  isirred2  17151  opprirred  17152
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