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Theorem un0mulcl 8962
Description: If  S is closed under multiplication, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
Hypotheses
Ref Expression
un0addcl.1  |-  ( ph  ->  S  C_  CC )
un0addcl.2  |-  T  =  ( S  u.  {
0 } )
un0mulcl.3  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  S )
Assertion
Ref Expression
un0mulcl  |-  ( (
ph  /\  ( M  e.  T  /\  N  e.  T ) )  -> 
( M  x.  N
)  e.  T )

Proof of Theorem un0mulcl
StepHypRef Expression
1 un0addcl.2 . . . . 5  |-  T  =  ( S  u.  {
0 } )
21eleq2i 2182 . . . 4  |-  ( N  e.  T  <->  N  e.  ( S  u.  { 0 } ) )
3 elun 3185 . . . 4  |-  ( N  e.  ( S  u.  { 0 } )  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
42, 3bitri 183 . . 3  |-  ( N  e.  T  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
51eleq2i 2182 . . . . . 6  |-  ( M  e.  T  <->  M  e.  ( S  u.  { 0 } ) )
6 elun 3185 . . . . . 6  |-  ( M  e.  ( S  u.  { 0 } )  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
75, 6bitri 183 . . . . 5  |-  ( M  e.  T  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
8 ssun1 3207 . . . . . . . . 9  |-  S  C_  ( S  u.  { 0 } )
98, 1sseqtrri 3100 . . . . . . . 8  |-  S  C_  T
10 un0mulcl.3 . . . . . . . 8  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  S )
119, 10sseldi 3063 . . . . . . 7  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  T )
1211expr 370 . . . . . 6  |-  ( (
ph  /\  M  e.  S )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
13 un0addcl.1 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  CC )
1413sselda 3065 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  S )  ->  N  e.  CC )
1514mul02d 8118 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  =  0 )
16 ssun2 3208 . . . . . . . . . . 11  |-  { 0 }  C_  ( S  u.  { 0 } )
1716, 1sseqtrri 3100 . . . . . . . . . 10  |-  { 0 }  C_  T
18 c0ex 7724 . . . . . . . . . . 11  |-  0  e.  _V
1918snss 3617 . . . . . . . . . 10  |-  ( 0  e.  T  <->  { 0 }  C_  T )
2017, 19mpbir 145 . . . . . . . . 9  |-  0  e.  T
2115, 20syl6eqel 2206 . . . . . . . 8  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  e.  T )
22 elsni 3513 . . . . . . . . . 10  |-  ( M  e.  { 0 }  ->  M  =  0 )
2322oveq1d 5755 . . . . . . . . 9  |-  ( M  e.  { 0 }  ->  ( M  x.  N )  =  ( 0  x.  N ) )
2423eleq1d 2184 . . . . . . . 8  |-  ( M  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( 0  x.  N )  e.  T
) )
2521, 24syl5ibrcom 156 . . . . . . 7  |-  ( (
ph  /\  N  e.  S )  ->  ( M  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
2625impancom 258 . . . . . 6  |-  ( (
ph  /\  M  e.  { 0 } )  -> 
( N  e.  S  ->  ( M  x.  N
)  e.  T ) )
2712, 26jaodan 769 . . . . 5  |-  ( (
ph  /\  ( M  e.  S  \/  M  e.  { 0 } ) )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
287, 27sylan2b 283 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
29 0cnd 7723 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
3029snssd 3633 . . . . . . . . . 10  |-  ( ph  ->  { 0 }  C_  CC )
3113, 30unssd 3220 . . . . . . . . 9  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
321, 31eqsstrid 3111 . . . . . . . 8  |-  ( ph  ->  T  C_  CC )
3332sselda 3065 . . . . . . 7  |-  ( (
ph  /\  M  e.  T )  ->  M  e.  CC )
3433mul01d 8119 . . . . . 6  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  =  0 )
3534, 20syl6eqel 2206 . . . . 5  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  e.  T )
36 elsni 3513 . . . . . . 7  |-  ( N  e.  { 0 }  ->  N  =  0 )
3736oveq2d 5756 . . . . . 6  |-  ( N  e.  { 0 }  ->  ( M  x.  N )  =  ( M  x.  0 ) )
3837eleq1d 2184 . . . . 5  |-  ( N  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( M  x.  0 )  e.  T
) )
3935, 38syl5ibrcom 156 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
4028, 39jaod 689 . . 3  |-  ( (
ph  /\  M  e.  T )  ->  (
( N  e.  S  \/  N  e.  { 0 } )  ->  ( M  x.  N )  e.  T ) )
414, 40syl5bi 151 . 2  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  T  ->  ( M  x.  N )  e.  T ) )
4241impr 374 1  |-  ( (
ph  /\  ( M  e.  T  /\  N  e.  T ) )  -> 
( M  x.  N
)  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314    e. wcel 1463    u. cun 3037    C_ wss 3039   {csn 3495  (class class class)co 5740   CCcc 7582   0cc0 7584    x. cmul 7589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420  ax-resscn 7676  ax-1cn 7677  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-sub 7899
This theorem is referenced by:  nn0mulcl  8964
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