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Theorem frlmval 27207
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 2966 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 2966 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 id 21 . . . . 5  |-  ( r  =  R  ->  r  =  R )
5 fveq2 5731 . . . . . . 7  |-  ( r  =  R  ->  (ringLMod `  r )  =  (ringLMod `  R ) )
65sneqd 3829 . . . . . 6  |-  ( r  =  R  ->  { (ringLMod `  r ) }  =  { (ringLMod `  R ) } )
76xpeq2d 4905 . . . . 5  |-  ( r  =  R  ->  (
i  X.  { (ringLMod `  r ) } )  =  ( i  X. 
{ (ringLMod `  R ) } ) )
84, 7oveq12d 6102 . . . 4  |-  ( r  =  R  ->  (
r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) )  =  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) ) )
9 xpeq1 4895 . . . . 5  |-  ( i  =  I  ->  (
i  X.  { (ringLMod `  R ) } )  =  ( I  X.  { (ringLMod `  R ) } ) )
109oveq2d 6100 . . . 4  |-  ( i  =  I  ->  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) )  =  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
11 df-frlm 27205 . . . 4  |- freeLMod  =  ( r  e.  _V , 
i  e.  _V  |->  ( r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) ) )
12 ovex 6109 . . . 4  |-  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) )  e.  _V
138, 10, 11, 12ovmpt2 6212 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R freeLMod  I )  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
142, 3, 13syl2an 465 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
151, 14syl5eq 2482 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    X. cxp 4879   ` cfv 5457  (class class class)co 6084  ringLModcrglmod 16246    (+)m cdsmm 27188   freeLMod cfrlm 27203
This theorem is referenced by:  frlmlmod  27208  frlmpws  27209  frlmlss  27210  frlmpwsfi  27211  frlmbas  27214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-frlm 27205
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