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Theorem irredmul 17484
Description: If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i  |-  I  =  (Irred `  R )
irredmul.b  |-  B  =  ( Base `  R
)
irredmul.u  |-  U  =  (Unit `  R )
irredmul.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
irredmul  |-  ( ( X  e.  B  /\  Y  e.  B  /\  ( X  .x.  Y )  e.  I )  -> 
( X  e.  U  \/  Y  e.  U
) )

Proof of Theorem irredmul
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irredmul.b . . . . 5  |-  B  =  ( Base `  R
)
2 irredmul.u . . . . 5  |-  U  =  (Unit `  R )
3 irredn0.i . . . . 5  |-  I  =  (Irred `  R )
4 irredmul.t . . . . 5  |-  .x.  =  ( .r `  R )
51, 2, 3, 4isirred2 17476 . . . 4  |-  ( ( X  .x.  Y )  e.  I  <->  ( ( X  .x.  Y )  e.  B  /\  -.  ( X  .x.  Y )  e.  U  /\  A. x  e.  B  A. y  e.  B  ( (
x  .x.  y )  =  ( X  .x.  Y )  ->  (
x  e.  U  \/  y  e.  U )
) ) )
65simp3bi 1013 . . 3  |-  ( ( X  .x.  Y )  e.  I  ->  A. x  e.  B  A. y  e.  B  ( (
x  .x.  y )  =  ( X  .x.  Y )  ->  (
x  e.  U  \/  y  e.  U )
) )
7 eqid 2457 . . . 4  |-  ( X 
.x.  Y )  =  ( X  .x.  Y
)
8 oveq1 6303 . . . . . . 7  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
98eqeq1d 2459 . . . . . 6  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  ( X 
.x.  Y )  <->  ( X  .x.  y )  =  ( X  .x.  Y ) ) )
10 eleq1 2529 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  U  <->  X  e.  U ) )
1110orbi1d 702 . . . . . 6  |-  ( x  =  X  ->  (
( x  e.  U  \/  y  e.  U
)  <->  ( X  e.  U  \/  y  e.  U ) ) )
129, 11imbi12d 320 . . . . 5  |-  ( x  =  X  ->  (
( ( x  .x.  y )  =  ( X  .x.  Y )  ->  ( x  e.  U  \/  y  e.  U ) )  <->  ( ( X  .x.  y )  =  ( X  .x.  Y
)  ->  ( X  e.  U  \/  y  e.  U ) ) ) )
13 oveq2 6304 . . . . . . 7  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1413eqeq1d 2459 . . . . . 6  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  ( X 
.x.  Y )  <->  ( X  .x.  Y )  =  ( X  .x.  Y ) ) )
15 eleq1 2529 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  U  <->  Y  e.  U ) )
1615orbi2d 701 . . . . . 6  |-  ( y  =  Y  ->  (
( X  e.  U  \/  y  e.  U
)  <->  ( X  e.  U  \/  Y  e.  U ) ) )
1714, 16imbi12d 320 . . . . 5  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  ( X  .x.  Y )  ->  ( X  e.  U  \/  y  e.  U ) )  <->  ( ( X  .x.  Y )  =  ( X  .x.  Y
)  ->  ( X  e.  U  \/  Y  e.  U ) ) ) )
1812, 17rspc2v 3219 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( ( x 
.x.  y )  =  ( X  .x.  Y
)  ->  ( x  e.  U  \/  y  e.  U ) )  -> 
( ( X  .x.  Y )  =  ( X  .x.  Y )  ->  ( X  e.  U  \/  Y  e.  U ) ) ) )
197, 18mpii 43 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( ( x 
.x.  y )  =  ( X  .x.  Y
)  ->  ( x  e.  U  \/  y  e.  U ) )  -> 
( X  e.  U  \/  Y  e.  U
) ) )
206, 19syl5 32 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  e.  I  ->  ( X  e.  U  \/  Y  e.  U
) ) )
21203impia 1193 1  |-  ( ( X  e.  B  /\  Y  e.  B  /\  ( X  .x.  Y )  e.  I )  -> 
( X  e.  U  \/  Y  e.  U
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   ` cfv 5594  (class class class)co 6296   Basecbs 14643   .rcmulr 14712  Unitcui 17414  Irredcir 17415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-irred 17418
This theorem is referenced by:  prmirredlem  18649  prmirredlemOLD  18652
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