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Theorem un0mulcl 8389
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 2146 . . . 4  |-  ( N  e.  T  <->  N  e.  ( S  u.  { 0 } ) )
3 elun 3114 . . . 4  |-  ( N  e.  ( S  u.  { 0 } )  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
42, 3bitri 182 . . 3  |-  ( N  e.  T  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
51eleq2i 2146 . . . . . 6  |-  ( M  e.  T  <->  M  e.  ( S  u.  { 0 } ) )
6 elun 3114 . . . . . 6  |-  ( M  e.  ( S  u.  { 0 } )  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
75, 6bitri 182 . . . . 5  |-  ( M  e.  T  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
8 ssun1 3136 . . . . . . . . 9  |-  S  C_  ( S  u.  { 0 } )
98, 1sseqtr4i 3033 . . . . . . . 8  |-  S  C_  T
10 un0mulcl.3 . . . . . . . 8  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  S )
119, 10sseldi 2998 . . . . . . 7  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  T )
1211expr 367 . . . . . 6  |-  ( (
ph  /\  M  e.  S )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
13 un0addcl.1 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  CC )
1413sselda 3000 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  S )  ->  N  e.  CC )
1514mul02d 7563 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  =  0 )
16 ssun2 3137 . . . . . . . . . . 11  |-  { 0 }  C_  ( S  u.  { 0 } )
1716, 1sseqtr4i 3033 . . . . . . . . . 10  |-  { 0 }  C_  T
18 c0ex 7175 . . . . . . . . . . 11  |-  0  e.  _V
1918snss 3524 . . . . . . . . . 10  |-  ( 0  e.  T  <->  { 0 }  C_  T )
2017, 19mpbir 144 . . . . . . . . 9  |-  0  e.  T
2115, 20syl6eqel 2170 . . . . . . . 8  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  e.  T )
22 elsni 3424 . . . . . . . . . 10  |-  ( M  e.  { 0 }  ->  M  =  0 )
2322oveq1d 5558 . . . . . . . . 9  |-  ( M  e.  { 0 }  ->  ( M  x.  N )  =  ( 0  x.  N ) )
2423eleq1d 2148 . . . . . . . 8  |-  ( M  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( 0  x.  N )  e.  T
) )
2521, 24syl5ibrcom 155 . . . . . . 7  |-  ( (
ph  /\  N  e.  S )  ->  ( M  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
2625impancom 256 . . . . . 6  |-  ( (
ph  /\  M  e.  { 0 } )  -> 
( N  e.  S  ->  ( M  x.  N
)  e.  T ) )
2712, 26jaodan 744 . . . . 5  |-  ( (
ph  /\  ( M  e.  S  \/  M  e.  { 0 } ) )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
287, 27sylan2b 281 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
29 0cnd 7174 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
3029snssd 3538 . . . . . . . . . 10  |-  ( ph  ->  { 0 }  C_  CC )
3113, 30unssd 3149 . . . . . . . . 9  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
321, 31syl5eqss 3044 . . . . . . . 8  |-  ( ph  ->  T  C_  CC )
3332sselda 3000 . . . . . . 7  |-  ( (
ph  /\  M  e.  T )  ->  M  e.  CC )
3433mul01d 7564 . . . . . 6  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  =  0 )
3534, 20syl6eqel 2170 . . . . 5  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  e.  T )
36 elsni 3424 . . . . . . 7  |-  ( N  e.  { 0 }  ->  N  =  0 )
3736oveq2d 5559 . . . . . 6  |-  ( N  e.  { 0 }  ->  ( M  x.  N )  =  ( M  x.  0 ) )
3837eleq1d 2148 . . . . 5  |-  ( N  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( M  x.  0 )  e.  T
) )
3935, 38syl5ibrcom 155 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
4028, 39jaod 670 . . 3  |-  ( (
ph  /\  M  e.  T )  ->  (
( N  e.  S  \/  N  e.  { 0 } )  ->  ( M  x.  N )  e.  T ) )
414, 40syl5bi 150 . 2  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  T  ->  ( M  x.  N )  e.  T ) )
4241impr 371 1  |-  ( (
ph  /\  ( M  e.  T  /\  N  e.  T ) )  -> 
( M  x.  N
)  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434    u. cun 2972    C_ wss 2974   {csn 3406  (class class class)co 5543   CCcc 7041   0cc0 7043    x. cmul 7048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348
This theorem is referenced by:  nn0mulcl  8391
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