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Theorem irredmul 15814
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 15806 . . . 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 974 . . 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 2436 . . . 4  |-  ( X 
.x.  Y )  =  ( X  .x.  Y
)
8 oveq1 6088 . . . . . . 7  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
98eqeq1d 2444 . . . . . 6  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  ( X 
.x.  Y )  <->  ( X  .x.  y )  =  ( X  .x.  Y ) ) )
10 eleq1 2496 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  U  <->  X  e.  U ) )
1110orbi1d 684 . . . . . 6  |-  ( x  =  X  ->  (
( x  e.  U  \/  y  e.  U
)  <->  ( X  e.  U  \/  y  e.  U ) ) )
129, 11imbi12d 312 . . . . 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 6089 . . . . . . 7  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1413eqeq1d 2444 . . . . . 6  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  ( X 
.x.  Y )  <->  ( X  .x.  Y )  =  ( X  .x.  Y ) ) )
15 eleq1 2496 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  U  <->  Y  e.  U ) )
1615orbi2d 683 . . . . . 6  |-  ( y  =  Y  ->  (
( X  e.  U  \/  y  e.  U
)  <->  ( X  e.  U  \/  Y  e.  U ) ) )
1714, 16imbi12d 312 . . . . 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 3058 . . . 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 41 . . 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 30 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  e.  I  ->  ( X  e.  U  \/  Y  e.  U
) ) )
21203impia 1150 1  |-  ( ( X  e.  B  /\  Y  e.  B  /\  ( X  .x.  Y )  e.  I )  -> 
( X  e.  U  \/  Y  e.  U
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469   .rcmulr 13530  Unitcui 15744  Irredcir 15745
This theorem is referenced by:  prmirredlem  16773
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-irred 15748
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