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Theorem frlmbas 27223
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 5539 . . . . 5  |-  (ringLMod `  R
)  e.  _V
2 fnconstg 5429 . . . . 5  |-  ( (ringLMod `  R )  e.  _V  ->  ( I  X.  {
(ringLMod `  R ) } )  Fn  I )
31, 2ax-mp 8 . . . 4  |-  ( I  X.  { (ringLMod `  R
) } )  Fn  I
4 eqid 2283 . . . . 5  |-  ( R
X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )
5 eqid 2283 . . . . 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 27203 . . . 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 651 . . 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 452 . 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 5594 . . . . . . . . . . . 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 651 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  ( 0g `  ( ( I  X.  { (ringLMod `  R ) } ) `  x
) ) )
1211adantl 452 . . . . . . . . . 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 5729 . . . . . . . . . . . . 13  |-  ( x  e.  I  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1413adantl 452 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1514fveq2d 5529 . . . . . . . . . . 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 15950 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  (ringLMod `  R ) )
1816, 17eqtri 2303 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  (ringLMod `  R
) )
1915, 18syl6eqr 2333 . . . . . . . . . 10  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  ( 0g `  ( ( I  X.  { (ringLMod `  R
) } ) `  x ) )  =  .0.  )
2012, 19eqtrd 2315 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  .0.  )
2120neeq2d 2460 . . . . . . . 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 2779 . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  ( Base `  R )  e.  _V
2523, 24eqeltri 2353 . . . . . . . . . . . 12  |-  N  e. 
_V
26 elmapg 6785 . . . . . . . . . . . 12  |-  ( ( N  e.  _V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2725, 26mpan 651 . . . . . . . . . . 11  |-  ( I  e.  W  ->  (
k  e.  ( N  ^m  I )  <->  k :
I --> N ) )
2827adantl 452 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2928biimpa 470 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k :
I --> N )
30 ffn 5389 . . . . . . . . 9  |-  ( k : I --> N  -> 
k  Fn  I )
3129, 30syl 15 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k  Fn  I )
32 fn0g 14385 . . . . . . . . 9  |-  0g  Fn  _V
33 ssv 3198 . . . . . . . . 9  |-  ran  (
I  X.  { (ringLMod `  R ) } ) 
C_  _V
34 fnco 5352 . . . . . . . . 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 1277 . . . . . . . 8  |-  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) )  Fn  I
36 fndmdif 5629 . . . . . . . 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 643 . . . . . . 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 5649 . . . . . . . 8  |-  ( k  Fn  I  ->  ( `' k " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( k `  x
)  =/=  .0.  }
)
3931, 38syl 15 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( `' k " ( _V  \  {  .0.  } ) )  =  { x  e.  I  |  ( k `
 x )  =/= 
.0.  } )
4022, 37, 393eqtr4d 2325 . . . . . 6  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  =  ( `' k " ( _V  \  {  .0.  }
) ) )
4140eleq1d 2349 . . . . 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 2779 . . . 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 2283 . . . . . . . . 9  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
44 rlmbas 15948 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
4523, 44eqtri 2303 . . . . . . . . 9  |-  N  =  ( Base `  (ringLMod `  R ) )
4643, 45pwsbas 13386 . . . . . . . 8  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
471, 46mpan 651 . . . . . . 7  |-  ( I  e.  W  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
4847adantl 452 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
49 eqid 2283 . . . . . . . . . . 11  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
5043, 49pwsval 13385 . . . . . . . . . 10  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
511, 50mpan 651 . . . . . . . . 9  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5251adantl 452 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
53 rlmsca 15952 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
5453adantr 451 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
5554oveq1d 5873 . . . . . . . 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 2318 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5756fveq2d 5529 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  (
(ringLMod `  R )  ^s  I
) )  =  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) ) )
5848, 57eqtrd 2315 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) ) )
59 rabeq 2782 . . . . 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 15 . . . 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 2317 . . 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 2327 . 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 27216 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
6564fveq2d 5529 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  F
)  =  ( Base `  ( R  (+)m  ( I  X.  { (ringLMod `  R
) } ) ) ) )
668, 62, 653eqtr4d 2325 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   Basecbs 13148  Scalarcsca 13211   X_scprds 13346    ^s cpws 13347   0gc0g 13400  ringLModcrglmod 15922    (+)m cdsmm 27197   freeLMod cfrlm 27212
This theorem is referenced by:  frlmelbas  27224  ellspd  27254  frlmpwfi  27262  islindf4  27308  matbas2  27475
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-sra 15925  df-rgmod 15926  df-dsmm 27198  df-frlm 27214
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