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Theorem un0mulcl 10254
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 2500 . . . 4  |-  ( N  e.  T  <->  N  e.  ( S  u.  { 0 } ) )
3 elun 3488 . . . 4  |-  ( N  e.  ( S  u.  { 0 } )  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
42, 3bitri 241 . . 3  |-  ( N  e.  T  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
51eleq2i 2500 . . . . . 6  |-  ( M  e.  T  <->  M  e.  ( S  u.  { 0 } ) )
6 elun 3488 . . . . . 6  |-  ( M  e.  ( S  u.  { 0 } )  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
75, 6bitri 241 . . . . 5  |-  ( M  e.  T  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
8 ssun1 3510 . . . . . . . . 9  |-  S  C_  ( S  u.  { 0 } )
98, 1sseqtr4i 3381 . . . . . . . 8  |-  S  C_  T
10 un0mulcl.3 . . . . . . . 8  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  S )
119, 10sseldi 3346 . . . . . . 7  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  T )
1211expr 599 . . . . . 6  |-  ( (
ph  /\  M  e.  S )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
13 un0addcl.1 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  CC )
1413sselda 3348 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  S )  ->  N  e.  CC )
1514mul02d 9264 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  =  0 )
16 ssun2 3511 . . . . . . . . . . 11  |-  { 0 }  C_  ( S  u.  { 0 } )
1716, 1sseqtr4i 3381 . . . . . . . . . 10  |-  { 0 }  C_  T
18 c0ex 9085 . . . . . . . . . . 11  |-  0  e.  _V
1918snss 3926 . . . . . . . . . 10  |-  ( 0  e.  T  <->  { 0 }  C_  T )
2017, 19mpbir 201 . . . . . . . . 9  |-  0  e.  T
2115, 20syl6eqel 2524 . . . . . . . 8  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  e.  T )
22 elsni 3838 . . . . . . . . . 10  |-  ( M  e.  { 0 }  ->  M  =  0 )
2322oveq1d 6096 . . . . . . . . 9  |-  ( M  e.  { 0 }  ->  ( M  x.  N )  =  ( 0  x.  N ) )
2423eleq1d 2502 . . . . . . . 8  |-  ( M  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( 0  x.  N )  e.  T
) )
2521, 24syl5ibrcom 214 . . . . . . 7  |-  ( (
ph  /\  N  e.  S )  ->  ( M  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
2625impancom 428 . . . . . 6  |-  ( (
ph  /\  M  e.  { 0 } )  -> 
( N  e.  S  ->  ( M  x.  N
)  e.  T ) )
2712, 26jaodan 761 . . . . 5  |-  ( (
ph  /\  ( M  e.  S  \/  M  e.  { 0 } ) )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
287, 27sylan2b 462 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
29 0cn 9084 . . . . . . . . . . . 12  |-  0  e.  CC
3029a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
3130snssd 3943 . . . . . . . . . 10  |-  ( ph  ->  { 0 }  C_  CC )
3213, 31unssd 3523 . . . . . . . . 9  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
331, 32syl5eqss 3392 . . . . . . . 8  |-  ( ph  ->  T  C_  CC )
3433sselda 3348 . . . . . . 7  |-  ( (
ph  /\  M  e.  T )  ->  M  e.  CC )
3534mul01d 9265 . . . . . 6  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  =  0 )
3635, 20syl6eqel 2524 . . . . 5  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  e.  T )
37 elsni 3838 . . . . . . 7  |-  ( N  e.  { 0 }  ->  N  =  0 )
3837oveq2d 6097 . . . . . 6  |-  ( N  e.  { 0 }  ->  ( M  x.  N )  =  ( M  x.  0 ) )
3938eleq1d 2502 . . . . 5  |-  ( N  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( M  x.  0 )  e.  T
) )
4036, 39syl5ibrcom 214 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
4128, 40jaod 370 . . 3  |-  ( (
ph  /\  M  e.  T )  ->  (
( N  e.  S  \/  N  e.  { 0 } )  ->  ( M  x.  N )  e.  T ) )
424, 41syl5bi 209 . 2  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  T  ->  ( M  x.  N )  e.  T ) )
4342impr 603 1  |-  ( (
ph  /\  ( M  e.  T  /\  N  e.  T ) )  -> 
( M  x.  N
)  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3318    C_ wss 3320   {csn 3814  (class class class)co 6081   CCcc 8988   0cc0 8990    x. cmul 8995
This theorem is referenced by:  nn0mulcl  10256  plymullem  20135
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  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2602  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-pw 3801  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-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125
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