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Theorem frlmval 18297
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 3085 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 3085 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 id 22 . . . . 5  |-  ( r  =  R  ->  r  =  R )
5 fveq2 5798 . . . . . . 7  |-  ( r  =  R  ->  (ringLMod `  r )  =  (ringLMod `  R ) )
65sneqd 3996 . . . . . 6  |-  ( r  =  R  ->  { (ringLMod `  r ) }  =  { (ringLMod `  R ) } )
76xpeq2d 4971 . . . . 5  |-  ( r  =  R  ->  (
i  X.  { (ringLMod `  r ) } )  =  ( i  X. 
{ (ringLMod `  R ) } ) )
84, 7oveq12d 6217 . . . 4  |-  ( r  =  R  ->  (
r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) )  =  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) ) )
9 xpeq1 4961 . . . . 5  |-  ( i  =  I  ->  (
i  X.  { (ringLMod `  R ) } )  =  ( I  X.  { (ringLMod `  R ) } ) )
109oveq2d 6215 . . . 4  |-  ( i  =  I  ->  ( R  (+)m  ( i  X.  {
(ringLMod `  R ) } ) )  =  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
11 df-frlm 18296 . . . 4  |- freeLMod  =  ( r  e.  _V , 
i  e.  _V  |->  ( r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) ) )
12 ovex 6224 . . . 4  |-  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) )  e.  _V
138, 10, 11, 12ovmpt2 6335 . . 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 2507 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 1370    e. wcel 1758   _Vcvv 3076   {csn 3984    X. cxp 4945   ` cfv 5525  (class class class)co 6199  ringLModcrglmod 17372    (+)m cdsmm 18280   freeLMod cfrlm 18295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-iota 5488  df-fun 5527  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-frlm 18296
This theorem is referenced by:  frlmlmod  18298  frlmpws  18299  frlmlss  18300  frlmpwsfi  18301  frlmbas  18304  frlmbasOLD  18305
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