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Theorem issubmd 27486
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 3271 . . 3  |-  { z  e.  B  |  ps }  C_  B
21a1i 10 . 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 14407 . . . 4  |-  ( M  e.  Mnd  ->  .0.  e.  B )
73, 6syl 15 . . 3  |-  ( ph  ->  .0.  e.  B )
8 issubmd.cz . . 3  |-  ( ph  ->  ch )
9 issubmd.ch . . . 4  |-  ( z  =  .0.  ->  ( ps 
<->  ch ) )
109elrab 2936 . . 3  |-  (  .0. 
e.  { z  e.  B  |  ps }  <->  (  .0.  e.  B  /\  ch ) )
117, 8, 10sylanbrc 645 . 2  |-  ( ph  ->  .0.  e.  { z  e.  B  |  ps } )
12 issubmd.th . . . . . 6  |-  ( z  =  x  ->  ( ps 
<->  th ) )
1312elrab 2936 . . . . 5  |-  ( x  e.  { z  e.  B  |  ps }  <->  ( x  e.  B  /\  th ) )
14 issubmd.ta . . . . . 6  |-  ( z  =  y  ->  ( ps 
<->  ta ) )
1514elrab 2936 . . . . 5  |-  ( y  e.  { z  e.  B  |  ps }  <->  ( y  e.  B  /\  ta ) )
1613, 15anbi12i 678 . . . 4  |-  ( ( x  e.  { z  e.  B  |  ps }  /\  y  e.  {
z  e.  B  |  ps } )  <->  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )
173adantr 451 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  M  e.  Mnd )
18 simprll 738 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  x  e.  B )
19 simprrl 740 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
y  e.  B )
20 issubmd.p . . . . . . 7  |-  .+  =  ( +g  `  M )
214, 20mndcl 14388 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
2217, 18, 19, 21syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  B )
23 an4 797 . . . . . 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 461 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  et )
26 issubmd.et . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( ps 
<->  et ) )
2726elrab 2936 . . . . 5  |-  ( ( x  .+  y )  e.  { z  e.  B  |  ps }  <->  ( ( x  .+  y
)  e.  B  /\  et ) )
2822, 25, 27sylanbrc 645 . . . 4  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  { z  e.  B  |  ps } )
2916, 28sylan2b 461 . . 3  |-  ( (
ph  /\  ( x  e.  { z  e.  B  |  ps }  /\  y  e.  { z  e.  B  |  ps } ) )  ->  ( x  .+  y )  e.  {
z  e.  B  |  ps } )
3029ralrimivva 2648 . 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 14441 . . 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 15 . 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 1135 1  |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377  SubMndcsubmnd 14430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-submnd 14432
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