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Theorem isirred 15794
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 5749 . . . 4  |-  ( X  e.  (Irred `  R
)  ->  R  e.  dom Irred )
2 irred.3 . . . 4  |-  I  =  (Irred `  R )
31, 2eleq2s 2527 . . 3  |-  ( X  e.  I  ->  R  e.  dom Irred )
4 elex 2956 . . 3  |-  ( R  e.  dom Irred  ->  R  e. 
_V )
53, 4syl 16 . 2  |-  ( X  e.  I  ->  R  e.  _V )
6 eldifi 3461 . . . . . 6  |-  ( X  e.  ( B  \  U )  ->  X  e.  B )
7 irred.4 . . . . . 6  |-  N  =  ( B  \  U
)
86, 7eleq2s 2527 . . . . 5  |-  ( X  e.  N  ->  X  e.  B )
9 irred.1 . . . . 5  |-  B  =  ( Base `  R
)
108, 9syl6eleq 2525 . . . 4  |-  ( X  e.  N  ->  X  e.  ( Base `  R
) )
1110elfvexd 5751 . . 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 5734 . . . . . . . 8  |-  ( Base `  r )  e.  _V
14 difexg 4343 . . . . . . . 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 5724 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  ( Base `  R
) )
1918, 9syl6eqr 2485 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( Base `  r )  =  B )
2017fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  (Unit `  R ) )
21 irred.2 . . . . . . . . . . . 12  |-  U  =  (Unit `  R )
2220, 21syl6eqr 2485 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (Unit `  r )  =  U )
2319, 22difeq12d 3458 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  ( B 
\  U ) )
2423, 7syl6eqr 2485 . . . . . . . . 9  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( Base `  r )  \  (Unit `  r )
)  =  N )
2516, 24eqtrd 2467 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  b  =  N )
2617fveq2d 5724 . . . . . . . . . . . . 13  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  =  ( .r `  R
) )
27 irred.5 . . . . . . . . . . . . 13  |-  .x.  =  ( .r `  R )
2826, 27syl6eqr 2485 . . . . . . . . . . . 12  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  ( .r `  r )  = 
.x.  )
2928oveqd 6090 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
x ( .r `  r ) y )  =  ( x  .x.  y ) )
3029neeq1d 2611 . . . . . . . . . 10  |-  ( ( r  =  R  /\  b  =  ( ( Base `  r )  \ 
(Unit `  r )
) )  ->  (
( x ( .r
`  r ) y )  =/=  z  <->  ( x  .x.  y )  =/=  z
) )
3125, 30raleqbidv 2908 . . . . . . . . 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 2908 . . . . . . . 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 2943 . . . . . . 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 3285 . . . . . 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 15738 . . . . . 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 5734 . . . . . . . . . 10  |-  ( Base `  R )  e.  _V
379, 36eqeltri 2505 . . . . . . . . 9  |-  B  e. 
_V
38 difexg 4343 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  U )  e. 
_V )
3937, 38ax-mp 8 . . . . . . . 8  |-  ( B 
\  U )  e. 
_V
407, 39eqeltri 2505 . . . . . . 7  |-  N  e. 
_V
4140rabex 4346 . . . . . 6  |-  { z  e.  N  |  A. x  e.  N  A. y  e.  N  (
x  .x.  y )  =/=  z }  e.  _V
4234, 35, 41fvmpt 5798 . . . . 5  |-  ( R  e.  _V  ->  (Irred `  R )  =  {
z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
432, 42syl5eq 2479 . . . 4  |-  ( R  e.  _V  ->  I  =  { z  e.  N  |  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/=  z } )
4443eleq2d 2502 . . 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 2607 . . . . 5  |-  ( z  =  X  ->  (
( x  .x.  y
)  =/=  z  <->  ( x  .x.  y )  =/=  X
) )
46452ralbidv 2739 . . . 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 3084 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701   _Vcvv 2948   [_csb 3243    \ cdif 3309   dom cdm 4870   ` cfv 5446  (class class class)co 6073   Basecbs 13459   .rcmulr 13520  Unitcui 15734  Irredcir 15735
This theorem is referenced by:  isnirred  15795  isirred2  15796  opprirred  15797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-irred 15738
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