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Theorem frlmval 19242
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 3096 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 3096 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 id 23 . . . . 5  |-  ( r  =  R  ->  r  =  R )
5 fveq2 5881 . . . . . . 7  |-  ( r  =  R  ->  (ringLMod `  r )  =  (ringLMod `  R ) )
65sneqd 4014 . . . . . 6  |-  ( r  =  R  ->  { (ringLMod `  r ) }  =  { (ringLMod `  R ) } )
76xpeq2d 4878 . . . . 5  |-  ( r  =  R  ->  (
i  X.  { (ringLMod `  r ) } )  =  ( i  X. 
{ (ringLMod `  R ) } ) )
84, 7oveq12d 6323 . . . 4  |-  ( r  =  R  ->  (
r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) )  =  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) ) )
9 xpeq1 4868 . . . . 5  |-  ( i  =  I  ->  (
i  X.  { (ringLMod `  R ) } )  =  ( I  X.  { (ringLMod `  R ) } ) )
109oveq2d 6321 . . . 4  |-  ( i  =  I  ->  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) )  =  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
11 df-frlm 19241 . . . 4  |- freeLMod  =  ( r  e.  _V , 
i  e.  _V  |->  ( r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) ) )
12 ovex 6333 . . . 4  |-  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) )  e.  _V
138, 10, 11, 12ovmpt2 6446 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R freeLMod  I )  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
142, 3, 13syl2an 479 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   {csn 4002    X. cxp 4852   ` cfv 5601  (class class class)co 6305  ringLModcrglmod 18327    (+)m cdsmm 19225   freeLMod cfrlm 19240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-frlm 19241
This theorem is referenced by:  frlmlmod  19243  frlmpws  19244  frlmlss  19245  frlmpwsfi  19246  frlmbas  19249
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