MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  islmim Unicode version

Theorem islmim 16097
Description: An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypotheses
Ref Expression
islmim.b  |-  B  =  ( Base `  R
)
islmim.c  |-  C  =  ( Base `  S
)
Assertion
Ref Expression
islmim  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )

Proof of Theorem islmim
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmim 16062 . . 3  |- LMIso  =  ( a  e.  LMod ,  b  e.  LMod  |->  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
2 ovex 6073 . . . 4  |-  ( a LMHom 
b )  e.  _V
32rabex 4322 . . 3  |-  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
4 oveq12 6057 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a LMHom  b )  =  ( R LMHom  S
) )
5 fveq2 5695 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
6 islmim.b . . . . . 6  |-  B  =  ( Base `  R
)
75, 6syl6eqr 2462 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
8 fveq2 5695 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
9 islmim.c . . . . . 6  |-  C  =  ( Base `  S
)
108, 9syl6eqr 2462 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
11 f1oeq23 5635 . . . . 5  |-  ( ( ( Base `  a
)  =  B  /\  ( Base `  b )  =  C )  ->  (
c : ( Base `  a ) -1-1-onto-> ( Base `  b
)  <->  c : B -1-1-onto-> C
) )
127, 10, 11syl2an 464 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( c : (
Base `  a ) -1-1-onto-> ( Base `  b )  <->  c : B
-1-1-onto-> C ) )
134, 12rabeqbidv 2919 . . 3  |-  ( ( a  =  R  /\  b  =  S )  ->  { c  e.  ( a LMHom  b )  |  c : ( Base `  a ) -1-1-onto-> ( Base `  b
) }  =  {
c  e.  ( R LMHom 
S )  |  c : B -1-1-onto-> C } )
141, 3, 13elovmpt2 6258 . 2  |-  ( F  e.  ( R LMIso  S
)  <->  ( R  e. 
LMod  /\  S  e.  LMod  /\  F  e.  { c  e.  ( R LMHom  S
)  |  c : B -1-1-onto-> C } ) )
15 df-3an 938 . 2  |-  ( ( R  e.  LMod  /\  S  e.  LMod  /\  F  e.  { c  e.  ( R LMHom 
S )  |  c : B -1-1-onto-> C } )  <->  ( ( R  e.  LMod  /\  S  e.  LMod )  /\  F  e.  { c  e.  ( R LMHom  S )  |  c : B -1-1-onto-> C }
) )
16 f1oeq1 5632 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
1716elrab 3060 . . . 4  |-  ( F  e.  { c  e.  ( R LMHom  S )  |  c : B -1-1-onto-> C } 
<->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C ) )
1817anbi2i 676 . . 3  |-  ( ( ( R  e.  LMod  /\  S  e.  LMod )  /\  F  e.  { c  e.  ( R LMHom  S
)  |  c : B -1-1-onto-> C } )  <->  ( ( R  e.  LMod  /\  S  e.  LMod )  /\  ( F  e.  ( R LMHom  S )  /\  F : B
-1-1-onto-> C ) ) )
19 lmhmlmod1 16072 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  R  e.  LMod )
20 lmhmlmod2 16071 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  S  e.  LMod )
2119, 20jca 519 . . . . 5  |-  ( F  e.  ( R LMHom  S
)  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2221adantr 452 . . . 4  |-  ( ( F  e.  ( R LMHom 
S )  /\  F : B -1-1-onto-> C )  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2322pm4.71ri 615 . . 3  |-  ( ( F  e.  ( R LMHom 
S )  /\  F : B -1-1-onto-> C )  <->  ( ( R  e.  LMod  /\  S  e.  LMod )  /\  ( F  e.  ( R LMHom  S )  /\  F : B
-1-1-onto-> C ) ) )
2418, 23bitr4i 244 . 2  |-  ( ( ( R  e.  LMod  /\  S  e.  LMod )  /\  F  e.  { c  e.  ( R LMHom  S
)  |  c : B -1-1-onto-> C } )  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )
2514, 15, 243bitri 263 1  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2678   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   Basecbs 13432   LModclmod 15913   LMHom clmhm 16058   LMIso clmim 16059
This theorem is referenced by:  lmimf1o  16098  lmimlmhm  16099  islmim2  16101  pwssplit4  27067  indlcim  27186
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-lmhm 16061  df-lmim 16062
  Copyright terms: Public domain W3C validator