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Theorem frlmval 27216
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 2796 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 2796 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 id 19 . . . . 5  |-  ( r  =  R  ->  r  =  R )
5 fveq2 5525 . . . . . . 7  |-  ( r  =  R  ->  (ringLMod `  r )  =  (ringLMod `  R ) )
65sneqd 3653 . . . . . 6  |-  ( r  =  R  ->  { (ringLMod `  r ) }  =  { (ringLMod `  R ) } )
76xpeq2d 4713 . . . . 5  |-  ( r  =  R  ->  (
i  X.  { (ringLMod `  r ) } )  =  ( i  X. 
{ (ringLMod `  R ) } ) )
84, 7oveq12d 5876 . . . 4  |-  ( r  =  R  ->  (
r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) )  =  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) ) )
9 xpeq1 4703 . . . . 5  |-  ( i  =  I  ->  (
i  X.  { (ringLMod `  R ) } )  =  ( I  X.  { (ringLMod `  R ) } ) )
109oveq2d 5874 . . . 4  |-  ( i  =  I  ->  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) )  =  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
11 df-frlm 27214 . . . 4  |- freeLMod  =  ( r  e.  _V , 
i  e.  _V  |->  ( r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) ) )
12 ovex 5883 . . . 4  |-  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) )  e.  _V
138, 10, 11, 12ovmpt2 5983 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R freeLMod  I )  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
142, 3, 13syl2an 463 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
151, 14syl5eq 2327 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858  ringLModcrglmod 15922    (+)m cdsmm 27197   freeLMod cfrlm 27212
This theorem is referenced by:  frlmlmod  27217  frlmpws  27218  frlmlss  27219  frlmpwsfi  27220  frlmbas  27223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-frlm 27214
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