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Theorem frlmbas 27200
Description: Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
frlmval.f  |-  F  =  ( R freeLMod  I )
frlmbas.n  |-  N  =  ( Base `  R
)
frlmbas.z  |-  .0.  =  ( 0g `  R )
frlmbas.b  |-  B  =  { k  e.  ( N  ^m  I )  |  ( `' k
" ( _V  \  {  .0.  } ) )  e.  Fin }
Assertion
Ref Expression
frlmbas  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  F ) )
Distinct variable groups:    k, N    R, k    k, I    k, W    k, V    .0. , k
Allowed substitution hints:    B( k)    F( k)

Proof of Theorem frlmbas
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvex 5742 . . . . 5  |-  (ringLMod `  R
)  e.  _V
2 fnconstg 5631 . . . . 5  |-  ( (ringLMod `  R )  e.  _V  ->  ( I  X.  {
(ringLMod `  R ) } )  Fn  I )
31, 2ax-mp 8 . . . 4  |-  ( I  X.  { (ringLMod `  R
) } )  Fn  I
4 eqid 2436 . . . . 5  |-  ( R
X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )
5 eqid 2436 . . . . 5  |-  { k  e.  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  |  dom  ( k  \ 
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }  =  { k  e.  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  |  dom  ( k  \ 
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }
64, 5dsmmbas2 27180 . . . 4  |-  ( ( ( I  X.  {
(ringLMod `  R ) } )  Fn  I  /\  I  e.  W )  ->  { k  e.  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) )  |  dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }  =  ( Base `  ( R  (+)m 
( I  X.  {
(ringLMod `  R ) } ) ) ) )
73, 6mpan 652 . . 3  |-  ( I  e.  W  ->  { k  e.  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  |  dom  ( k  \ 
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }  =  ( Base `  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) ) )
87adantl 453 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { k  e.  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) )  |  dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }  =  ( Base `  ( R  (+)m 
( I  X.  {
(ringLMod `  R ) } ) ) ) )
9 frlmbas.b . . 3  |-  B  =  { k  e.  ( N  ^m  I )  |  ( `' k
" ( _V  \  {  .0.  } ) )  e.  Fin }
10 fvco2 5798 . . . . . . . . . . . 12  |-  ( ( ( I  X.  {
(ringLMod `  R ) } )  Fn  I  /\  x  e.  I )  ->  ( ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) `  x )  =  ( 0g `  ( ( I  X.  { (ringLMod `  R ) } ) `  x
) ) )
113, 10mpan 652 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  ( 0g `  ( ( I  X.  { (ringLMod `  R ) } ) `  x
) ) )
1211adantl 453 . . . . . . . . . 10  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  ( 0g `  ( ( I  X.  { (ringLMod `  R ) } ) `  x
) ) )
131fvconst2 5947 . . . . . . . . . . . . 13  |-  ( x  e.  I  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1413adantl 453 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1514fveq2d 5732 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  ( 0g `  ( ( I  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( 0g `  (ringLMod `  R ) ) )
16 frlmbas.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  R )
17 rlm0 16269 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  (ringLMod `  R ) )
1816, 17eqtri 2456 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  (ringLMod `  R
) )
1915, 18syl6eqr 2486 . . . . . . . . . 10  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  ( 0g `  ( ( I  X.  { (ringLMod `  R
) } ) `  x ) )  =  .0.  )
2012, 19eqtrd 2468 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  .0.  )
2120neeq2d 2615 . . . . . . . 8  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( k `  x
)  =/=  ( ( 0g  o.  ( I  X.  { (ringLMod `  R
) } ) ) `
 x )  <->  ( k `  x )  =/=  .0.  ) )
2221rabbidva 2947 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  { x  e.  I  |  (
k `  x )  =/=  ( ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) `  x ) }  =  { x  e.  I  |  (
k `  x )  =/=  .0.  } )
23 frlmbas.n . . . . . . . . . . . . 13  |-  N  =  ( Base `  R
)
24 fvex 5742 . . . . . . . . . . . . 13  |-  ( Base `  R )  e.  _V
2523, 24eqeltri 2506 . . . . . . . . . . . 12  |-  N  e. 
_V
26 elmapg 7031 . . . . . . . . . . . 12  |-  ( ( N  e.  _V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2725, 26mpan 652 . . . . . . . . . . 11  |-  ( I  e.  W  ->  (
k  e.  ( N  ^m  I )  <->  k :
I --> N ) )
2827adantl 453 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2928biimpa 471 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k :
I --> N )
30 ffn 5591 . . . . . . . . 9  |-  ( k : I --> N  -> 
k  Fn  I )
3129, 30syl 16 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k  Fn  I )
32 fn0g 14708 . . . . . . . . 9  |-  0g  Fn  _V
33 ssv 3368 . . . . . . . . 9  |-  ran  (
I  X.  { (ringLMod `  R ) } ) 
C_  _V
34 fnco 5553 . . . . . . . . 9  |-  ( ( 0g  Fn  _V  /\  ( I  X.  { (ringLMod `  R ) } )  Fn  I  /\  ran  ( I  X.  { (ringLMod `  R ) } ) 
C_  _V )  ->  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) )  Fn  I )
3532, 3, 33, 34mp3an 1279 . . . . . . . 8  |-  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) )  Fn  I
36 fndmdif 5834 . . . . . . . 8  |-  ( ( k  Fn  I  /\  ( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) )  Fn  I )  ->  dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  =  {
x  e.  I  |  ( k `  x
)  =/=  ( ( 0g  o.  ( I  X.  { (ringLMod `  R
) } ) ) `
 x ) } )
3731, 35, 36sylancl 644 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  =  {
x  e.  I  |  ( k `  x
)  =/=  ( ( 0g  o.  ( I  X.  { (ringLMod `  R
) } ) ) `
 x ) } )
38 fnniniseg2 5854 . . . . . . . 8  |-  ( k  Fn  I  ->  ( `' k " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( k `  x
)  =/=  .0.  }
)
3931, 38syl 16 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( `' k " ( _V  \  {  .0.  } ) )  =  { x  e.  I  |  ( k `
 x )  =/= 
.0.  } )
4022, 37, 393eqtr4d 2478 . . . . . 6  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  =  ( `' k " ( _V  \  {  .0.  }
) ) )
4140eleq1d 2502 . . . . 5  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin  <->  ( `' k " ( _V  \  {  .0.  } ) )  e.  Fin ) )
4241rabbidva 2947 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { k  e.  ( N  ^m  I )  |  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }  =  { k  e.  ( N  ^m  I
)  |  ( `' k " ( _V 
\  {  .0.  }
) )  e.  Fin } )
43 eqid 2436 . . . . . . . . 9  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
44 rlmbas 16267 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
4523, 44eqtri 2456 . . . . . . . . 9  |-  N  =  ( Base `  (ringLMod `  R ) )
4643, 45pwsbas 13709 . . . . . . . 8  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
471, 46mpan 652 . . . . . . 7  |-  ( I  e.  W  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
4847adantl 453 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
49 eqid 2436 . . . . . . . . . . 11  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
5043, 49pwsval 13708 . . . . . . . . . 10  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
511, 50mpan 652 . . . . . . . . 9  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5251adantl 453 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
53 rlmsca 16271 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
5453adantr 452 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
5554oveq1d 6096 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( (Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5652, 55eqtr4d 2471 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5756fveq2d 5732 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  (
(ringLMod `  R )  ^s  I
) )  =  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) ) )
5848, 57eqtrd 2468 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) ) )
59 rabeq 2950 . . . . 5  |-  ( ( N  ^m  I )  =  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  ->  { k  e.  ( N  ^m  I )  |  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }  =  { k  e.  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  |  dom  ( k  \ 
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin } )
6058, 59syl 16 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { k  e.  ( N  ^m  I )  |  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin }  =  { k  e.  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  |  dom  ( k  \ 
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin } )
6142, 60eqtr3d 2470 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { k  e.  ( N  ^m  I )  |  ( `' k
" ( _V  \  {  .0.  } ) )  e.  Fin }  =  { k  e.  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) )  |  dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin } )
629, 61syl5eq 2480 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  { k  e.  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  |  dom  ( k  \ 
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin } )
63 frlmval.f . . . 4  |-  F  =  ( R freeLMod  I )
6463frlmval 27193 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
6564fveq2d 5732 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  F
)  =  ( Base `  ( R  (+)m  ( I  X.  { (ringLMod `  R
) } ) ) ) )
668, 62, 653eqtr4d 2478 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   {csn 3814    X. cxp 4876   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881    o. ccom 4882    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   Basecbs 13469  Scalarcsca 13532   X_scprds 13669    ^s cpws 13670   0gc0g 13723  ringLModcrglmod 16241    (+)m cdsmm 27174   freeLMod cfrlm 27189
This theorem is referenced by:  frlmelbas  27201  ellspd  27231  frlmpwfi  27239  islindf4  27285  matbas2  27452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-sra 16244  df-rgmod 16245  df-dsmm 27175  df-frlm 27191
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