MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frlmbasOLD Structured version   Unicode version

Theorem frlmbasOLD 18181
Description: Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) Obsolete version of frlmbas 18180 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 5701 . . . . 5  |-  (ringLMod `  R
)  e.  _V
2 fnconstg 5598 . . . . 5  |-  ( (ringLMod `  R )  e.  _V  ->  ( I  X.  {
(ringLMod `  R ) } )  Fn  I )
31, 2ax-mp 5 . . . 4  |-  ( I  X.  { (ringLMod `  R
) } )  Fn  I
4 eqid 2443 . . . . 5  |-  ( R
X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )
5 eqid 2443 . . . . 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 18162 . . . 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 670 . . 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 466 . 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 5766 . . . . . . . . . . . 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 670 . . . . . . . . . . 11  |-  ( x  e.  I  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  ( 0g `  ( ( I  X.  { (ringLMod `  R ) } ) `  x
) ) )
1211adantl 466 . . . . . . . . . 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 5933 . . . . . . . . . . . . 13  |-  ( x  e.  I  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1413adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1514fveq2d 5695 . . . . . . . . . . 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 17278 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  (ringLMod `  R ) )
1816, 17eqtri 2463 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  (ringLMod `  R
) )
1915, 18syl6eqr 2493 . . . . . . . . . 10  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  ( 0g `  ( ( I  X.  { (ringLMod `  R
) } ) `  x ) )  =  .0.  )
2012, 19eqtrd 2475 . . . . . . . . 9  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  .0.  )
2120neeq2d 2622 . . . . . . . 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 2963 . . . . . . 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 5701 . . . . . . . . . . . . 13  |-  ( Base `  R )  e.  _V
2523, 24eqeltri 2513 . . . . . . . . . . . 12  |-  N  e. 
_V
26 elmapg 7227 . . . . . . . . . . . 12  |-  ( ( N  e.  _V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2725, 26mpan 670 . . . . . . . . . . 11  |-  ( I  e.  W  ->  (
k  e.  ( N  ^m  I )  <->  k :
I --> N ) )
2827adantl 466 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( k  e.  ( N  ^m  I )  <-> 
k : I --> N ) )
2928biimpa 484 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k :
I --> N )
30 ffn 5559 . . . . . . . . 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 15433 . . . . . . . . 9  |-  0g  Fn  _V
33 ssv 3376 . . . . . . . . 9  |-  ran  (
I  X.  { (ringLMod `  R ) } ) 
C_  _V
34 fnco 5519 . . . . . . . . 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 1314 . . . . . . . 8  |-  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) )  Fn  I
36 fndmdif 5807 . . . . . . . 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 662 . . . . . . 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 5827 . . . . . . . 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 2485 . . . . . 6  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  =  ( `' k " ( _V  \  {  .0.  }
) ) )
4140eleq1d 2509 . . . . 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 2963 . . . 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 2443 . . . . . . . . 9  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
44 rlmbas 17276 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
4523, 44eqtri 2463 . . . . . . . . 9  |-  N  =  ( Base `  (ringLMod `  R ) )
4643, 45pwsbas 14425 . . . . . . . 8  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
471, 46mpan 670 . . . . . . 7  |-  ( I  e.  W  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
4847adantl 466 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
49 eqid 2443 . . . . . . . . . . 11  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
5043, 49pwsval 14424 . . . . . . . . . 10  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
511, 50mpan 670 . . . . . . . . 9  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5251adantl 466 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
53 rlmsca 17281 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
5453adantr 465 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
5554oveq1d 6106 . . . . . . . 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 2478 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5756fveq2d 5695 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  (
(ringLMod `  R )  ^s  I
) )  =  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) ) )
5848, 57eqtrd 2475 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) ) )
59 rabeq 2966 . . . . 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 2477 . . 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 2487 . 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 18173 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
6564fveq2d 5695 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  F
)  =  ( Base `  ( R  (+)m  ( I  X.  { (ringLMod `  R
) } ) ) ) )
668, 62, 653eqtr4d 2485 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   {crab 2719   _Vcvv 2972    \ cdif 3325    C_ wss 3328   {csn 3877    X. cxp 4838   `'ccnv 4839   dom cdm 4840   ran crn 4841   "cima 4843    o. ccom 4844    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   Fincfn 7310   Basecbs 14174  Scalarcsca 14241   0gc0g 14378   X_scprds 14384    ^s cpws 14385  ringLModcrglmod 17250    (+)m cdsmm 18156   freeLMod cfrlm 18171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-prds 14386  df-pws 14388  df-sra 17253  df-rgmod 17254  df-dsmm 18157  df-frlm 18172
This theorem is referenced by:  frlmelbasOLD  18183  ellspdOLD  18231  frlmpwfi  29453
  Copyright terms: Public domain W3C validator