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Theorem issubmd 27052
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 3371 . . 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 14641 . . . 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 3035 . . 3  |-  (  .0. 
e.  { z  e.  B  |  ps }  <->  (  .0.  e.  B  /\  ch ) )
117, 8, 10sylanbrc 646 . 2  |-  ( ph  ->  .0.  e.  { z  e.  B  |  ps } )
12 issubmd.th . . . . . 6  |-  ( z  =  x  ->  ( ps 
<->  th ) )
1312elrab 3035 . . . . 5  |-  ( x  e.  { z  e.  B  |  ps }  <->  ( x  e.  B  /\  th ) )
14 issubmd.ta . . . . . 6  |-  ( z  =  y  ->  ( ps 
<->  ta ) )
1514elrab 3035 . . . . 5  |-  ( y  e.  { z  e.  B  |  ps }  <->  ( y  e.  B  /\  ta ) )
1613, 15anbi12i 679 . . . 4  |-  ( ( x  e.  { z  e.  B  |  ps }  /\  y  e.  {
z  e.  B  |  ps } )  <->  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )
173adantr 452 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  M  e.  Mnd )
18 simprll 739 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  x  e.  B )
19 simprrl 741 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
y  e.  B )
20 issubmd.p . . . . . . 7  |-  .+  =  ( +g  `  M )
214, 20mndcl 14622 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
2217, 18, 19, 21syl3anc 1184 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  B )
23 an4 798 . . . . . 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 462 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  et )
26 issubmd.et . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( ps 
<->  et ) )
2726elrab 3035 . . . . 5  |-  ( ( x  .+  y )  e.  { z  e.  B  |  ps }  <->  ( ( x  .+  y
)  e.  B  /\  et ) )
2822, 25, 27sylanbrc 646 . . . 4  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  { z  e.  B  |  ps } )
2916, 28sylan2b 462 . . 3  |-  ( (
ph  /\  ( x  e.  { z  e.  B  |  ps }  /\  y  e.  { z  e.  B  |  ps } ) )  ->  ( x  .+  y )  e.  {
z  e.  B  |  ps } )
3029ralrimivva 2741 . 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 14675 . . 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 1137 1  |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653    C_ wss 3263   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   0gc0g 13650   Mndcmnd 14611  SubMndcsubmnd 14664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-riota 6485  df-0g 13654  df-mnd 14617  df-submnd 14666
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