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Theorem frlmval 18574
Description: Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypothesis
Ref Expression
frlmval.f  |-  F  =  ( R freeLMod  I )
Assertion
Ref Expression
frlmval  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )

Proof of Theorem frlmval
Dummy variables  r 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmval.f . 2  |-  F  =  ( R freeLMod  I )
2 elex 3122 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 3122 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 id 22 . . . . 5  |-  ( r  =  R  ->  r  =  R )
5 fveq2 5866 . . . . . . 7  |-  ( r  =  R  ->  (ringLMod `  r )  =  (ringLMod `  R ) )
65sneqd 4039 . . . . . 6  |-  ( r  =  R  ->  { (ringLMod `  r ) }  =  { (ringLMod `  R ) } )
76xpeq2d 5023 . . . . 5  |-  ( r  =  R  ->  (
i  X.  { (ringLMod `  r ) } )  =  ( i  X. 
{ (ringLMod `  R ) } ) )
84, 7oveq12d 6302 . . . 4  |-  ( r  =  R  ->  (
r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) )  =  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) ) )
9 xpeq1 5013 . . . . 5  |-  ( i  =  I  ->  (
i  X.  { (ringLMod `  R ) } )  =  ( I  X.  { (ringLMod `  R ) } ) )
109oveq2d 6300 . . . 4  |-  ( i  =  I  ->  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) )  =  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
11 df-frlm 18573 . . . 4  |- freeLMod  =  ( r  e.  _V , 
i  e.  _V  |->  ( r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) ) )
12 ovex 6309 . . . 4  |-  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) )  e.  _V
138, 10, 11, 12ovmpt2 6422 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R freeLMod  I )  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
142, 3, 13syl2an 477 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
151, 14syl5eq 2520 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   ` cfv 5588  (class class class)co 6284  ringLModcrglmod 17615    (+)m cdsmm 18557   freeLMod cfrlm 18572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-frlm 18573
This theorem is referenced by:  frlmlmod  18575  frlmpws  18576  frlmlss  18577  frlmpwsfi  18578  frlmbas  18581  frlmbasOLD  18582
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