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Theorem islmim 17148
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 17109 . . 3  |- LMIso  =  ( a  e.  LMod ,  b  e.  LMod  |->  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) } )
2 ovex 6121 . . . 4  |-  ( a LMHom 
b )  e.  _V
32rabex 4448 . . 3  |-  { c  e.  ( a LMHom  b
)  |  c : ( Base `  a
)
-1-1-onto-> ( Base `  b ) }  e.  _V
4 oveq12 6105 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( a LMHom  b )  =  ( R LMHom  S
) )
5 fveq2 5696 . . . . . 6  |-  ( a  =  R  ->  ( Base `  a )  =  ( Base `  R
) )
6 islmim.b . . . . . 6  |-  B  =  ( Base `  R
)
75, 6syl6eqr 2493 . . . . 5  |-  ( a  =  R  ->  ( Base `  a )  =  B )
8 fveq2 5696 . . . . . 6  |-  ( b  =  S  ->  ( Base `  b )  =  ( Base `  S
) )
9 islmim.c . . . . . 6  |-  C  =  ( Base `  S
)
108, 9syl6eqr 2493 . . . . 5  |-  ( b  =  S  ->  ( Base `  b )  =  C )
11 f1oeq23 5640 . . . . 5  |-  ( ( ( Base `  a
)  =  B  /\  ( Base `  b )  =  C )  ->  (
c : ( Base `  a ) -1-1-onto-> ( Base `  b
)  <->  c : B -1-1-onto-> C
) )
127, 10, 11syl2an 477 . . . 4  |-  ( ( a  =  R  /\  b  =  S )  ->  ( c : (
Base `  a ) -1-1-onto-> ( Base `  b )  <->  c : B
-1-1-onto-> C ) )
134, 12rabeqbidv 2972 . . 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 6312 . 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 967 . 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 5637 . . . . 5  |-  ( c  =  F  ->  (
c : B -1-1-onto-> C  <->  F : B
-1-1-onto-> C ) )
1716elrab 3122 . . . 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 694 . . 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 17119 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  R  e.  LMod )
20 lmhmlmod2 17118 . . . . . 6  |-  ( F  e.  ( R LMHom  S
)  ->  S  e.  LMod )
2119, 20jca 532 . . . . 5  |-  ( F  e.  ( R LMHom  S
)  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2221adantr 465 . . . 4  |-  ( ( F  e.  ( R LMHom 
S )  /\  F : B -1-1-onto-> C )  ->  ( R  e.  LMod  /\  S  e.  LMod ) )
2322pm4.71ri 633 . . 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 252 . 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 271 1  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2724   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   Basecbs 14179   LModclmod 16953   LMHom clmhm 17105   LMIso clmim 17106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-lmhm 17108  df-lmim 17109
This theorem is referenced by:  lmimf1o  17149  lmimlmhm  17150  islmim2  17152  indlcim  18274  lmimco  18278  pwssplit4  29447
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