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Theorem un0mulcl 10830
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 2545 . . . 4  |-  ( N  e.  T  <->  N  e.  ( S  u.  { 0 } ) )
3 elun 3645 . . . 4  |-  ( N  e.  ( S  u.  { 0 } )  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
42, 3bitri 249 . . 3  |-  ( N  e.  T  <->  ( N  e.  S  \/  N  e.  { 0 } ) )
51eleq2i 2545 . . . . . 6  |-  ( M  e.  T  <->  M  e.  ( S  u.  { 0 } ) )
6 elun 3645 . . . . . 6  |-  ( M  e.  ( S  u.  { 0 } )  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
75, 6bitri 249 . . . . 5  |-  ( M  e.  T  <->  ( M  e.  S  \/  M  e.  { 0 } ) )
8 ssun1 3667 . . . . . . . . 9  |-  S  C_  ( S  u.  { 0 } )
98, 1sseqtr4i 3537 . . . . . . . 8  |-  S  C_  T
10 un0mulcl.3 . . . . . . . 8  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  S )
119, 10sseldi 3502 . . . . . . 7  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  T )
1211expr 615 . . . . . 6  |-  ( (
ph  /\  M  e.  S )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
13 un0addcl.1 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  CC )
1413sselda 3504 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  S )  ->  N  e.  CC )
1514mul02d 9777 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  =  0 )
16 ssun2 3668 . . . . . . . . . . 11  |-  { 0 }  C_  ( S  u.  { 0 } )
1716, 1sseqtr4i 3537 . . . . . . . . . 10  |-  { 0 }  C_  T
18 c0ex 9590 . . . . . . . . . . 11  |-  0  e.  _V
1918snss 4151 . . . . . . . . . 10  |-  ( 0  e.  T  <->  { 0 }  C_  T )
2017, 19mpbir 209 . . . . . . . . 9  |-  0  e.  T
2115, 20syl6eqel 2563 . . . . . . . 8  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  e.  T )
22 elsni 4052 . . . . . . . . . 10  |-  ( M  e.  { 0 }  ->  M  =  0 )
2322oveq1d 6299 . . . . . . . . 9  |-  ( M  e.  { 0 }  ->  ( M  x.  N )  =  ( 0  x.  N ) )
2423eleq1d 2536 . . . . . . . 8  |-  ( M  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( 0  x.  N )  e.  T
) )
2521, 24syl5ibrcom 222 . . . . . . 7  |-  ( (
ph  /\  N  e.  S )  ->  ( M  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
2625impancom 440 . . . . . 6  |-  ( (
ph  /\  M  e.  { 0 } )  -> 
( N  e.  S  ->  ( M  x.  N
)  e.  T ) )
2712, 26jaodan 783 . . . . 5  |-  ( (
ph  /\  ( M  e.  S  \/  M  e.  { 0 } ) )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
287, 27sylan2b 475 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  S  ->  ( M  x.  N )  e.  T ) )
29 0cnd 9589 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
3029snssd 4172 . . . . . . . . . 10  |-  ( ph  ->  { 0 }  C_  CC )
3113, 30unssd 3680 . . . . . . . . 9  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
321, 31syl5eqss 3548 . . . . . . . 8  |-  ( ph  ->  T  C_  CC )
3332sselda 3504 . . . . . . 7  |-  ( (
ph  /\  M  e.  T )  ->  M  e.  CC )
3433mul01d 9778 . . . . . 6  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  =  0 )
3534, 20syl6eqel 2563 . . . . 5  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  e.  T )
36 elsni 4052 . . . . . . 7  |-  ( N  e.  { 0 }  ->  N  =  0 )
3736oveq2d 6300 . . . . . 6  |-  ( N  e.  { 0 }  ->  ( M  x.  N )  =  ( M  x.  0 ) )
3837eleq1d 2536 . . . . 5  |-  ( N  e.  { 0 }  ->  ( ( M  x.  N )  e.  T  <->  ( M  x.  0 )  e.  T
) )
3935, 38syl5ibrcom 222 . . . 4  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  { 0 }  ->  ( M  x.  N )  e.  T
) )
4028, 39jaod 380 . . 3  |-  ( (
ph  /\  M  e.  T )  ->  (
( N  e.  S  \/  N  e.  { 0 } )  ->  ( M  x.  N )  e.  T ) )
414, 40syl5bi 217 . 2  |-  ( (
ph  /\  M  e.  T )  ->  ( N  e.  T  ->  ( M  x.  N )  e.  T ) )
4241impr 619 1  |-  ( (
ph  /\  ( M  e.  T  /\  N  e.  T ) )  -> 
( M  x.  N
)  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    C_ wss 3476   {csn 4027  (class class class)co 6284   CCcc 9490   0cc0 9492    x. cmul 9497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633
This theorem is referenced by:  nn0mulcl  10832  plymullem  22376
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