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Theorem irredmul 15507
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 15499 . . . 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 972 . . 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 2296 . . . 4  |-  ( X 
.x.  Y )  =  ( X  .x.  Y
)
8 oveq1 5881 . . . . . . 7  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
98eqeq1d 2304 . . . . . 6  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  ( X 
.x.  Y )  <->  ( X  .x.  y )  =  ( X  .x.  Y ) ) )
10 eleq1 2356 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  U  <->  X  e.  U ) )
1110orbi1d 683 . . . . . 6  |-  ( x  =  X  ->  (
( x  e.  U  \/  y  e.  U
)  <->  ( X  e.  U  \/  y  e.  U ) ) )
129, 11imbi12d 311 . . . . 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 5882 . . . . . . 7  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
1413eqeq1d 2304 . . . . . 6  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  ( X 
.x.  Y )  <->  ( X  .x.  Y )  =  ( X  .x.  Y ) ) )
15 eleq1 2356 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  U  <->  Y  e.  U ) )
1615orbi2d 682 . . . . . 6  |-  ( y  =  Y  ->  (
( X  e.  U  \/  y  e.  U
)  <->  ( X  e.  U  \/  Y  e.  U ) ) )
1714, 16imbi12d 311 . . . . 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 2903 . . . 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 39 . . 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 28 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  e.  I  ->  ( X  e.  U  \/  Y  e.  U
) ) )
21203impia 1148 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 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225  Unitcui 15437  Irredcir 15438
This theorem is referenced by:  prmirredlem  16462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-irred 15441
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