Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  issubmd Unicode version

Theorem issubmd 26736
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 3219 . . 3  |-  { z  e.  B  |  ps }  C_  B
21a1i 12 . 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 14339 . . . 4  |-  ( M  e.  Mnd  ->  .0.  e.  B )
73, 6syl 17 . . 3  |-  ( ph  ->  .0.  e.  B )
8 issubmd.cz . . 3  |-  ( ph  ->  ch )
9 issubmd.ch . . . 4  |-  ( z  =  .0.  ->  ( ps 
<->  ch ) )
109elrab 2891 . . 3  |-  (  .0. 
e.  { z  e.  B  |  ps }  <->  (  .0.  e.  B  /\  ch ) )
117, 8, 10sylanbrc 648 . 2  |-  ( ph  ->  .0.  e.  { z  e.  B  |  ps } )
12 issubmd.th . . . . . 6  |-  ( z  =  x  ->  ( ps 
<->  th ) )
1312elrab 2891 . . . . 5  |-  ( x  e.  { z  e.  B  |  ps }  <->  ( x  e.  B  /\  th ) )
14 issubmd.ta . . . . . 6  |-  ( z  =  y  ->  ( ps 
<->  ta ) )
1514elrab 2891 . . . . 5  |-  ( y  e.  { z  e.  B  |  ps }  <->  ( y  e.  B  /\  ta ) )
1613, 15anbi12i 681 . . . 4  |-  ( ( x  e.  { z  e.  B  |  ps }  /\  y  e.  {
z  e.  B  |  ps } )  <->  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )
173adantr 453 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  M  e.  Mnd )
18 simprll 741 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  x  e.  B )
19 simprrl 743 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
y  e.  B )
20 issubmd.p . . . . . . 7  |-  .+  =  ( +g  `  M )
214, 20mndcl 14320 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
2217, 18, 19, 21syl3anc 1187 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  B )
23 an4 800 . . . . . 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 463 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  et )
26 issubmd.et . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( ps 
<->  et ) )
2726elrab 2891 . . . . 5  |-  ( ( x  .+  y )  e.  { z  e.  B  |  ps }  <->  ( ( x  .+  y
)  e.  B  /\  et ) )
2822, 25, 27sylanbrc 648 . . . 4  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  { z  e.  B  |  ps } )
2916, 28sylan2b 463 . . 3  |-  ( (
ph  /\  ( x  e.  { z  e.  B  |  ps }  /\  y  e.  { z  e.  B  |  ps } ) )  ->  ( x  .+  y )  e.  {
z  e.  B  |  ps } )
3029ralrimivva 2608 . 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 14373 . . 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 17 . 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 1140 1  |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2516   {crab 2520    C_ wss 3113   ` cfv 4659  (class class class)co 5778   Basecbs 13096   +g cplusg 13156   0gc0g 13348   Mndcmnd 14309  SubMndcsubmnd 14362
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-iota 6211  df-riota 6258  df-0g 13352  df-mnd 14315  df-submnd 14364
  Copyright terms: Public domain W3C validator