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Theorem isirred 15759
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 5716 . . . 4  |-  ( X  e.  (Irred `  R
)  ->  R  e.  dom Irred )
2 irred.3 . . . 4  |-  I  =  (Irred `  R )
31, 2eleq2s 2496 . . 3  |-  ( X  e.  I  ->  R  e.  dom Irred )
4 elex 2924 . . 3  |-  ( R  e.  dom Irred  ->  R  e. 
_V )
53, 4syl 16 . 2  |-  ( X  e.  I  ->  R  e.  _V )
6 eldifi 3429 . . . . . 6  |-  ( X  e.  ( B  \  U )  ->  X  e.  B )
7 irred.4 . . . . . 6  |-  N  =  ( B  \  U
)
86, 7eleq2s 2496 . . . . 5  |-  ( X  e.  N  ->  X  e.  B )
9 irred.1 . . . . 5  |-  B  =  ( Base `  R
)
108, 9syl6eleq 2494 . . . 4  |-  ( X  e.  N  ->  X  e.  ( Base `  R
) )
1110elfvexd 5718 . . 3  |-  ( X  e.  N  ->  R  e.  _V )
1211adantr 452 . 2  |-  ( ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  X )  ->  R  e.  _V )
13 fvex 5701 . . . . . . . 8  |-  ( Base `  r )  e.  _V
14 difexg 4311 . . . . . . . 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 448 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  b  =  ( ( Base `  r )  \  (Unit `  r ) ) )
17 simpl 444 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  r  =  R )
1817fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  ( Base `  R
) )
1918, 9syl6eqr 2454 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  B )
2017fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  (Unit `  R ) )
21 irred.2 . . . . . . . . . . . 12  |-  U  =  (Unit `  R )
2220, 21syl6eqr 2454 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  U )
2319, 22difeq12d 3426 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  ( B 
\  U ) )
2423, 7syl6eqr 2454 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  N )
2516, 24eqtrd 2436 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  b  =  N )
2617fveq2d 5691 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  =  ( .r `  R
) )
27 irred.5 . . . . . . . . . . . . 13  |-  .x.  =  ( .r `  R )
2826, 27syl6eqr 2454 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  = 
.x.  )
2928oveqd 6057 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
3029neeq1d 2580 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( x ( .r
`  r ) y )  =/=  z  <->  ( x  .x.  y )  =/=  z
) )
3125, 30raleqbidv 2876 . . . . . . . . 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 2876 . . . . . . . 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 2911 . . . . . . 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 3253 . . . . . 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 15703 . . . . . 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 5701 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
379, 36eqeltri 2474 . . . . . . . . 9  |-  B  e. 
_V
38 difexg 4311 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  U )  e. 
_V )
3937, 38ax-mp 8 . . . . . . . 8  |-  ( B 
\  U )  e. 
_V
407, 39eqeltri 2474 . . . . . . 7  |-  N  e. 
_V
4140rabex 4314 . . . . . 6  |-  { z  e.  N  |  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  z }  e.  _V
4234, 35, 41fvmpt 5765 . . . . 5  |-  ( R  e.  _V  ->  (Irred `  R )  =  {
z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
432, 42syl5eq 2448 . . . 4  |-  ( R  e.  _V  ->  I  =  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
4443eleq2d 2471 . . 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 2576 . . . . 5  |-  ( z  =  X  ->  (
( x  .x.  y
)  =/=  z  <->  ( x  .x.  y )  =/=  X
) )
46452ralbidv 2708 . . . 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 3052 . . 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 253 . 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 343 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916   [_csb 3211    \ cdif 3277   dom cdm 4837   ` cfv 5413  (class class class)co 6040   Basecbs 13424   .rcmulr 13485  Unitcui 15699  Irredcir 15700
This theorem is referenced by:  isnirred  15760  isirred2  15761  opprirred  15762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-irred 15703
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