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Theorem un0mulcl 10851
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 2535 . . . 4  |-  ( N  e.  T  <->  N  e.  ( S  u.  { 0 } ) )
3 elun 3641 . . . 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 2535 . . . . . 6  |-  ( M  e.  T  <->  M  e.  ( S  u.  { 0 } ) )
6 elun 3641 . . . . . 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 3663 . . . . . . . . 9  |-  S  C_  ( S  u.  { 0 } )
98, 1sseqtr4i 3532 . . . . . . . 8  |-  S  C_  T
10 un0mulcl.3 . . . . . . . 8  |-  ( (
ph  /\  ( M  e.  S  /\  N  e.  S ) )  -> 
( M  x.  N
)  e.  S )
119, 10sseldi 3497 . . . . . . 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 3499 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  S )  ->  N  e.  CC )
1514mul02d 9795 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  =  0 )
16 ssun2 3664 . . . . . . . . . . 11  |-  { 0 }  C_  ( S  u.  { 0 } )
1716, 1sseqtr4i 3532 . . . . . . . . . 10  |-  { 0 }  C_  T
18 c0ex 9607 . . . . . . . . . . 11  |-  0  e.  _V
1918snss 4156 . . . . . . . . . 10  |-  ( 0  e.  T  <->  { 0 }  C_  T )
2017, 19mpbir 209 . . . . . . . . 9  |-  0  e.  T
2115, 20syl6eqel 2553 . . . . . . . 8  |-  ( (
ph  /\  N  e.  S )  ->  (
0  x.  N )  e.  T )
22 elsni 4057 . . . . . . . . . 10  |-  ( M  e.  { 0 }  ->  M  =  0 )
2322oveq1d 6311 . . . . . . . . 9  |-  ( M  e.  { 0 }  ->  ( M  x.  N )  =  ( 0  x.  N ) )
2423eleq1d 2526 . . . . . . . 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 785 . . . . 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 9606 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
3029snssd 4177 . . . . . . . . . 10  |-  ( ph  ->  { 0 }  C_  CC )
3113, 30unssd 3676 . . . . . . . . 9  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
321, 31syl5eqss 3543 . . . . . . . 8  |-  ( ph  ->  T  C_  CC )
3332sselda 3499 . . . . . . 7  |-  ( (
ph  /\  M  e.  T )  ->  M  e.  CC )
3433mul01d 9796 . . . . . 6  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  =  0 )
3534, 20syl6eqel 2553 . . . . 5  |-  ( (
ph  /\  M  e.  T )  ->  ( M  x.  0 )  e.  T )
36 elsni 4057 . . . . . . 7  |-  ( N  e.  { 0 }  ->  N  =  0 )
3736oveq2d 6312 . . . . . 6  |-  ( N  e.  { 0 }  ->  ( M  x.  N )  =  ( M  x.  0 ) )
3837eleq1d 2526 . . . . 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 1395    e. wcel 1819    u. cun 3469    C_ wss 3471   {csn 4032  (class class class)co 6296   CCcc 9507   0cc0 9509    x. cmul 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650
This theorem is referenced by:  nn0mulcl  10853  plymullem  22738
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