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Theorem issubmd 15579
Description: Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
issubmd.b  |-  B  =  ( Base `  M
)
issubmd.p  |-  .+  =  ( +g  `  M )
issubmd.z  |-  .0.  =  ( 0g `  M )
issubmd.m  |-  ( ph  ->  M  e.  Mnd )
issubmd.cz  |-  ( ph  ->  ch )
issubmd.cp  |-  ( (
ph  /\  ( (
x  e.  B  /\  y  e.  B )  /\  ( th  /\  ta ) ) )  ->  et )
issubmd.ch  |-  ( z  =  .0.  ->  ( ps 
<->  ch ) )
issubmd.th  |-  ( z  =  x  ->  ( ps 
<->  th ) )
issubmd.ta  |-  ( z  =  y  ->  ( ps 
<->  ta ) )
issubmd.et  |-  ( z  =  ( x  .+  y )  ->  ( ps 
<->  et ) )
Assertion
Ref Expression
issubmd  |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
Distinct variable groups:    x, y,
z, B    x, M, y    ph, x, y    ps, x, y    z,  .+    z,  .0.    ch, z    et, z    ta, z    th, z
Allowed substitution hints:    ph( z)    ps( z)    ch( x, y)    th( x, y)    ta( x, y)    et( x, y)    .+ ( x, y)    M( z)    .0. ( x, y)

Proof of Theorem issubmd
StepHypRef Expression
1 ssrab2 3535 . . 3  |-  { z  e.  B  |  ps }  C_  B
21a1i 11 . 2  |-  ( ph  ->  { z  e.  B  |  ps }  C_  B
)
3 issubmd.m . . . 4  |-  ( ph  ->  M  e.  Mnd )
4 issubmd.b . . . . 5  |-  B  =  ( Base `  M
)
5 issubmd.z . . . . 5  |-  .0.  =  ( 0g `  M )
64, 5mndidcl 15541 . . . 4  |-  ( M  e.  Mnd  ->  .0.  e.  B )
73, 6syl 16 . . 3  |-  ( ph  ->  .0.  e.  B )
8 issubmd.cz . . 3  |-  ( ph  ->  ch )
9 issubmd.ch . . . 4  |-  ( z  =  .0.  ->  ( ps 
<->  ch ) )
109elrab 3214 . . 3  |-  (  .0. 
e.  { z  e.  B  |  ps }  <->  (  .0.  e.  B  /\  ch ) )
117, 8, 10sylanbrc 664 . 2  |-  ( ph  ->  .0.  e.  { z  e.  B  |  ps } )
12 issubmd.th . . . . . 6  |-  ( z  =  x  ->  ( ps 
<->  th ) )
1312elrab 3214 . . . . 5  |-  ( x  e.  { z  e.  B  |  ps }  <->  ( x  e.  B  /\  th ) )
14 issubmd.ta . . . . . 6  |-  ( z  =  y  ->  ( ps 
<->  ta ) )
1514elrab 3214 . . . . 5  |-  ( y  e.  { z  e.  B  |  ps }  <->  ( y  e.  B  /\  ta ) )
1613, 15anbi12i 697 . . . 4  |-  ( ( x  e.  { z  e.  B  |  ps }  /\  y  e.  {
z  e.  B  |  ps } )  <->  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )
173adantr 465 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  M  e.  Mnd )
18 simprll 761 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  x  e.  B )
19 simprrl 763 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
y  e.  B )
20 issubmd.p . . . . . . 7  |-  .+  =  ( +g  `  M )
214, 20mndcl 15522 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
2217, 18, 19, 21syl3anc 1219 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  B )
23 an4 820 . . . . . 6  |-  ( ( ( x  e.  B  /\  th )  /\  (
y  e.  B  /\  ta ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  ( th  /\  ta ) ) )
24 issubmd.cp . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  y  e.  B )  /\  ( th  /\  ta ) ) )  ->  et )
2523, 24sylan2b 475 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  et )
26 issubmd.et . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( ps 
<->  et ) )
2726elrab 3214 . . . . 5  |-  ( ( x  .+  y )  e.  { z  e.  B  |  ps }  <->  ( ( x  .+  y
)  e.  B  /\  et ) )
2822, 25, 27sylanbrc 664 . . . 4  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  { z  e.  B  |  ps } )
2916, 28sylan2b 475 . . 3  |-  ( (
ph  /\  ( x  e.  { z  e.  B  |  ps }  /\  y  e.  { z  e.  B  |  ps } ) )  ->  ( x  .+  y )  e.  {
z  e.  B  |  ps } )
3029ralrimivva 2904 . 2  |-  ( ph  ->  A. x  e.  {
z  e.  B  |  ps } A. y  e. 
{ z  e.  B  |  ps }  ( x 
.+  y )  e. 
{ z  e.  B  |  ps } )
314, 5, 20issubm 15577 . . 3  |-  ( M  e.  Mnd  ->  ( { z  e.  B  |  ps }  e.  (SubMnd `  M )  <->  ( {
z  e.  B  |  ps }  C_  B  /\  .0.  e.  { z  e.  B  |  ps }  /\  A. x  e.  {
z  e.  B  |  ps } A. y  e. 
{ z  e.  B  |  ps }  ( x 
.+  y )  e. 
{ z  e.  B  |  ps } ) ) )
323, 31syl 16 . 2  |-  ( ph  ->  ( { z  e.  B  |  ps }  e.  (SubMnd `  M )  <->  ( { z  e.  B  |  ps }  C_  B  /\  .0.  e.  { z  e.  B  |  ps }  /\  A. x  e. 
{ z  e.  B  |  ps } A. y  e.  { z  e.  B  |  ps }  ( x 
.+  y )  e. 
{ z  e.  B  |  ps } ) ) )
332, 11, 30, 32mpbir3and 1171 1  |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799    C_ wss 3426   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   0gc0g 14480   Mndcmnd 15511  SubMndcsubmnd 15565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-iota 5479  df-fun 5518  df-fv 5524  df-riota 6151  df-ov 6193  df-0g 14482  df-mnd 15517  df-submnd 15567
This theorem is referenced by:  mrcmndind  15596
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