ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  issubmd Unicode version

Theorem issubmd 12742
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 3240 . . 3  |-  { z  e.  B  |  ps }  C_  B
21a1i 9 . 2  |-  ( ph  ->  { z  e.  B  |  ps }  C_  B
)
3 issubmd.ch . . 3  |-  ( z  =  .0.  ->  ( ps 
<->  ch ) )
4 issubmd.m . . . 4  |-  ( ph  ->  M  e.  Mnd )
5 issubmd.b . . . . 5  |-  B  =  ( Base `  M
)
6 issubmd.z . . . . 5  |-  .0.  =  ( 0g `  M )
75, 6mndidcl 12710 . . . 4  |-  ( M  e.  Mnd  ->  .0.  e.  B )
84, 7syl 14 . . 3  |-  ( ph  ->  .0.  e.  B )
9 issubmd.cz . . 3  |-  ( ph  ->  ch )
103, 8, 9elrabd 2895 . 2  |-  ( ph  ->  .0.  e.  { z  e.  B  |  ps } )
11 issubmd.th . . . . . 6  |-  ( z  =  x  ->  ( ps 
<->  th ) )
1211elrab 2893 . . . . 5  |-  ( x  e.  { z  e.  B  |  ps }  <->  ( x  e.  B  /\  th ) )
13 issubmd.ta . . . . . 6  |-  ( z  =  y  ->  ( ps 
<->  ta ) )
1413elrab 2893 . . . . 5  |-  ( y  e.  { z  e.  B  |  ps }  <->  ( y  e.  B  /\  ta ) )
1512, 14anbi12i 460 . . . 4  |-  ( ( x  e.  { z  e.  B  |  ps }  /\  y  e.  {
z  e.  B  |  ps } )  <->  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )
16 issubmd.et . . . . 5  |-  ( z  =  ( x  .+  y )  ->  ( ps 
<->  et ) )
174adantr 276 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  M  e.  Mnd )
18 simprll 537 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  x  e.  B )
19 simprrl 539 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
y  e.  B )
20 issubmd.p . . . . . . 7  |-  .+  =  ( +g  `  M )
215, 20mndcl 12703 . . . . . 6  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
2217, 18, 19, 21syl3anc 1238 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  B )
23 an4 586 . . . . . 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 287 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  ->  et )
2616, 22, 25elrabd 2895 . . . 4  |-  ( (
ph  /\  ( (
x  e.  B  /\  th )  /\  ( y  e.  B  /\  ta ) ) )  -> 
( x  .+  y
)  e.  { z  e.  B  |  ps } )
2715, 26sylan2b 287 . . 3  |-  ( (
ph  /\  ( x  e.  { z  e.  B  |  ps }  /\  y  e.  { z  e.  B  |  ps } ) )  ->  ( x  .+  y )  e.  {
z  e.  B  |  ps } )
2827ralrimivva 2559 . 2  |-  ( ph  ->  A. x  e.  {
z  e.  B  |  ps } A. y  e. 
{ z  e.  B  |  ps }  ( x 
.+  y )  e. 
{ z  e.  B  |  ps } )
295, 6, 20issubm 12740 . . 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 } ) ) )
304, 29syl 14 . 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 } ) ) )
312, 10, 28, 30mpbir3and 1180 1  |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459    C_ wss 3129   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640   Mndcmnd 12696  SubMndcsubmnd 12727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-submnd 12729
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator