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Theorem frlmbasOLD 19083
Description: Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) Obsolete version of frlmbas 19082 as of 23-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frlmval.f  |-  F  =  ( R freeLMod  I )
frlmbasOLD.n  |-  N  =  ( Base `  R
)
frlmbasOLD.z  |-  .0.  =  ( 0g `  R )
frlmbasOLD.b  |-  B  =  { k  e.  ( N  ^m  I )  |  ( `' k
" ( _V  \  {  .0.  } ) )  e.  Fin }
Assertion
Ref Expression
frlmbasOLD  |-  ( ( 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 frlmbasOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvex 5858 . . . . 5  |-  (ringLMod `  R
)  e.  _V
2 fnconstg 5755 . . . . 5  |-  ( (ringLMod `  R )  e.  _V  ->  ( I  X.  {
(ringLMod `  R ) } )  Fn  I )
31, 2ax-mp 5 . . . 4  |-  ( I  X.  { (ringLMod `  R
) } )  Fn  I
4 eqid 2402 . . . . 5  |-  ( R
X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )
5 eqid 2402 . . . . 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 19064 . . . 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 668 . . 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 464 . 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 frlmbasOLD.b . . 3  |-  B  =  { k  e.  ( N  ^m  I )  |  ( `' k
" ( _V  \  {  .0.  } ) )  e.  Fin }
10 fvco2 5923 . . . . . . . . . . . 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 668 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  ( 0g `  ( ( I  X.  { (ringLMod `  R ) } ) `  x
) ) )
1211adantl 464 . . . . . . . . . 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 6106 . . . . . . . . . . . . 13  |-  ( x  e.  I  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1413adantl 464 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1514fveq2d 5852 . . . . . . . . . . 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 frlmbasOLD.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  R )
17 rlm0 18161 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  (ringLMod `  R ) )
1816, 17eqtri 2431 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  (ringLMod `  R
) )
1915, 18syl6eqr 2461 . . . . . . . . . 10  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  ( 0g `  ( ( I  X.  { (ringLMod `  R
) } ) `  x ) )  =  .0.  )
2012, 19eqtrd 2443 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  .0.  )
2120neeq2d 2681 . . . . . . . 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 3049 . . . . . . 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 frlmbasOLD.n . . . . . . . . . . . . 13  |-  N  =  ( Base `  R
)
24 fvex 5858 . . . . . . . . . . . . 13  |-  ( Base `  R )  e.  _V
2523, 24eqeltri 2486 . . . . . . . . . . . 12  |-  N  e. 
_V
26 elmapg 7469 . . . . . . . . . . . 12  |-  ( ( N  e.  _V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2725, 26mpan 668 . . . . . . . . . . 11  |-  ( I  e.  W  ->  (
k  e.  ( N  ^m  I )  <->  k :
I --> N ) )
2827adantl 464 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2928biimpa 482 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k :
I --> N )
30 ffn 5713 . . . . . . . . 9  |-  ( k : I --> N  -> 
k  Fn  I )
3129, 30syl 17 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k  Fn  I )
32 fn0g 16211 . . . . . . . . 9  |-  0g  Fn  _V
33 ssv 3461 . . . . . . . . 9  |-  ran  (
I  X.  { (ringLMod `  R ) } ) 
C_  _V
34 fnco 5669 . . . . . . . . 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 1326 . . . . . . . 8  |-  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) )  Fn  I
36 fndmdif 5968 . . . . . . . 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 660 . . . . . . 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 fnniniseg2OLD 5988 . . . . . . . 8  |-  ( k  Fn  I  ->  ( `' k " ( _V  \  {  .0.  }
) )  =  {
x  e.  I  |  ( k `  x
)  =/=  .0.  }
)
3931, 38syl 17 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( `' k " ( _V  \  {  .0.  } ) )  =  { x  e.  I  |  ( k `
 x )  =/= 
.0.  } )
4022, 37, 393eqtr4d 2453 . . . . . 6  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  =  ( `' k " ( _V  \  {  .0.  }
) ) )
4140eleq1d 2471 . . . . 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 3049 . . . 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 2402 . . . . . . . . 9  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
44 rlmbas 18159 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
4523, 44eqtri 2431 . . . . . . . . 9  |-  N  =  ( Base `  (ringLMod `  R ) )
4643, 45pwsbas 15099 . . . . . . . 8  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
471, 46mpan 668 . . . . . . 7  |-  ( I  e.  W  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
4847adantl 464 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
49 eqid 2402 . . . . . . . . . . 11  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
5043, 49pwsval 15098 . . . . . . . . . 10  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
511, 50mpan 668 . . . . . . . . 9  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5251adantl 464 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
53 rlmsca 18164 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
5453adantr 463 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
5554oveq1d 6292 . . . . . . . 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 2446 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5756fveq2d 5852 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  (
(ringLMod `  R )  ^s  I
) )  =  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) ) )
5848, 57eqtrd 2443 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) ) )
59 rabeq 3052 . . . . 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 17 . . . 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 2445 . . 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 2455 . 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 19075 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
6564fveq2d 5852 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  F
)  =  ( Base `  ( R  (+)m  ( I  X.  { (ringLMod `  R
) } ) ) ) )
668, 62, 653eqtr4d 2453 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2757   _Vcvv 3058    \ cdif 3410    C_ wss 3413   {csn 3971    X. cxp 4820   `'ccnv 4821   dom cdm 4822   ran crn 4823   "cima 4825    o. ccom 4826    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277    ^m cmap 7456   Fincfn 7553   Basecbs 14839  Scalarcsca 14910   0gc0g 15052   X_scprds 15058    ^s cpws 15059  ringLModcrglmod 18133    (+)m cdsmm 19058   freeLMod cfrlm 19073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-fz 11725  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-hom 14931  df-cco 14932  df-0g 15054  df-prds 15060  df-pws 15062  df-sra 18136  df-rgmod 18137  df-dsmm 19059  df-frlm 19074
This theorem is referenced by: (None)
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