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Theorem frlmbas 18546
Description: Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
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 finSupp  .0.  }
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 5867 . . . . 5  |-  (ringLMod `  R
)  e.  _V
2 fnconstg 5764 . . . . 5  |-  ( (ringLMod `  R )  e.  _V  ->  ( I  X.  {
(ringLMod `  R ) } )  Fn  I )
31, 2ax-mp 5 . . . 4  |-  ( I  X.  { (ringLMod `  R
) } )  Fn  I
4 eqid 2460 . . . . 5  |-  ( R
X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )
5 eqid 2460 . . . . 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 18528 . . . 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 frlmbas.b . . 3  |-  B  =  { k  e.  ( N  ^m  I )  |  k finSupp  .0.  }
10 fvco2 5933 . . . . . . . . . . . . 13  |-  ( ( ( 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 . . . . . . . . . . . 12  |-  ( x  e.  I  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  ( 0g `  ( ( I  X.  { (ringLMod `  R ) } ) `  x
) ) )
1211adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( 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 6107 . . . . . . . . . . . . . 14  |-  ( x  e.  I  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1413adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  x )  =  (ringLMod `  R
) )
1514fveq2d 5861 . . . . . . . . . . . 12  |-  ( ( ( ( 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 . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  R )
17 rlm0 17619 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  (ringLMod `  R ) )
1816, 17eqtri 2489 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  (ringLMod `  R
) )
1915, 18syl6eqr 2519 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  ( 0g `  ( ( I  X.  { (ringLMod `  R
) } ) `  x ) )  =  .0.  )
2012, 19eqtrd 2501 . . . . . . . . . 10  |-  ( ( ( ( R  e.  V  /\  I  e.  W )  /\  k  e.  ( N  ^m  I
) )  /\  x  e.  I )  ->  (
( 0g  o.  (
I  X.  { (ringLMod `  R ) } ) ) `  x )  =  .0.  )
2120neeq2d 2738 . . . . . . . . 9  |-  ( ( ( ( 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 3097 . . . . . . . 8  |-  ( ( ( 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 elmapfn 7431 . . . . . . . . . 10  |-  ( k  e.  ( N  ^m  I )  ->  k  Fn  I )
2423adantl 466 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  k  Fn  I )
25 fn0g 15739 . . . . . . . . . 10  |-  0g  Fn  _V
26 ssv 3517 . . . . . . . . . 10  |-  ran  (
I  X.  { (ringLMod `  R ) } ) 
C_  _V
27 fnco 5680 . . . . . . . . . 10  |-  ( ( 0g  Fn  _V  /\  ( I  X.  { (ringLMod `  R ) } )  Fn  I  /\  ran  ( I  X.  { (ringLMod `  R ) } ) 
C_  _V )  ->  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) )  Fn  I )
2825, 3, 26, 27mp3an 1319 . . . . . . . . 9  |-  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) )  Fn  I
29 fndmdif 5976 . . . . . . . . 9  |-  ( ( 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 ) } )
3024, 28, 29sylancl 662 . . . . . . . 8  |-  ( ( ( 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 ) } )
31 simplr 754 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  I  e.  W )
32 fvex 5867 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
3316, 32eqeltri 2544 . . . . . . . . . 10  |-  .0.  e.  _V
3433a1i 11 . . . . . . . . 9  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  .0.  e.  _V )
35 suppvalfn 6898 . . . . . . . . 9  |-  ( ( k  Fn  I  /\  I  e.  W  /\  .0.  e.  _V )  -> 
( k supp  .0.  )  =  { x  e.  I  |  ( k `  x )  =/=  .0.  } )
3624, 31, 34, 35syl3anc 1223 . . . . . . . 8  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( k supp  .0.  )  =  {
x  e.  I  |  ( k `  x
)  =/=  .0.  }
)
3722, 30, 363eqtr4d 2511 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  dom  ( k 
\  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  =  ( k supp  .0.  ) )
3837eleq1d 2529 . . . . . 6  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin  <->  ( k supp  .0.  )  e.  Fin ) )
39 elmapfun 7432 . . . . . . . . 9  |-  ( k  e.  ( N  ^m  I )  ->  Fun  k )
40 id 22 . . . . . . . . 9  |-  ( k  e.  ( N  ^m  I )  ->  k  e.  ( N  ^m  I
) )
4133a1i 11 . . . . . . . . 9  |-  ( k  e.  ( N  ^m  I )  ->  .0.  e.  _V )
4239, 40, 413jca 1171 . . . . . . . 8  |-  ( k  e.  ( N  ^m  I )  ->  ( Fun  k  /\  k  e.  ( N  ^m  I
)  /\  .0.  e.  _V ) )
4342adantl 466 . . . . . . 7  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( Fun  k  /\  k  e.  ( N  ^m  I )  /\  .0.  e.  _V ) )
44 funisfsupp 7823 . . . . . . 7  |-  ( ( Fun  k  /\  k  e.  ( N  ^m  I
)  /\  .0.  e.  _V )  ->  ( k finSupp  .0. 
<->  ( k supp  .0.  )  e.  Fin ) )
4543, 44syl 16 . . . . . 6  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( k finSupp  .0.  <->  ( k supp  .0.  )  e.  Fin ) )
4638, 45bitr4d 256 . . . . 5  |-  ( ( ( R  e.  V  /\  I  e.  W
)  /\  k  e.  ( N  ^m  I ) )  ->  ( dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin  <->  k finSupp  .0.  )
)
4746rabbidva 3097 . . . 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 finSupp  .0.  } )
48 eqid 2460 . . . . . . . . 9  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
49 frlmbas.n . . . . . . . . . 10  |-  N  =  ( Base `  R
)
50 rlmbas 17617 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
5149, 50eqtri 2489 . . . . . . . . 9  |-  N  =  ( Base `  (ringLMod `  R ) )
5248, 51pwsbas 14731 . . . . . . . 8  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
531, 52mpan 670 . . . . . . 7  |-  ( I  e.  W  ->  ( N  ^m  I )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) ) )
5453adantl 466 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
55 eqid 2460 . . . . . . . . . . 11  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
5648, 55pwsval 14730 . . . . . . . . . 10  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
571, 56mpan 670 . . . . . . . . 9  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
5857adantl 466 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
59 rlmsca 17622 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
6059adantr 465 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
6160oveq1d 6290 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( (Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
6258, 61eqtr4d 2504 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
6362fveq2d 5861 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  (
(ringLMod `  R )  ^s  I
) )  =  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) ) )
6454, 63eqtrd 2501 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( N  ^m  I
)  =  ( Base `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) ) )
65 rabeq 3100 . . . . 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 } )
6664, 65syl 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 } )
6747, 66eqtr3d 2503 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  { k  e.  ( N  ^m  I )  |  k finSupp  .0.  }  =  { k  e.  (
Base `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) )  |  dom  ( k  \  ( 0g  o.  ( I  X.  { (ringLMod `  R ) } ) ) )  e.  Fin } )
689, 67syl5eq 2513 . 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 } )
69 frlmval.f . . . 4  |-  F  =  ( R freeLMod  I )
7069frlmval 18539 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R 
(+)m  ( I  X.  {
(ringLMod `  R ) } ) ) )
7170fveq2d 5861 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  F
)  =  ( Base `  ( R  (+)m  ( I  X.  { (ringLMod `  R
) } ) ) ) )
728, 68, 713eqtr4d 2511 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  ( Base `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   {crab 2811   _Vcvv 3106    \ cdif 3466    C_ wss 3469   {csn 4020   class class class wbr 4440    X. cxp 4990   dom cdm 4992   ran crn 4993    o. ccom 4996   Fun wfun 5573    Fn wfn 5574   ` cfv 5579  (class class class)co 6275   supp csupp 6891    ^m cmap 7410   Fincfn 7506   finSupp cfsupp 7818   Basecbs 14479  Scalarcsca 14547   0gc0g 14684   X_scprds 14690    ^s cpws 14691  ringLModcrglmod 17591    (+)m cdsmm 18522   freeLMod cfrlm 18537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-hom 14568  df-cco 14569  df-0g 14686  df-prds 14692  df-pws 14694  df-sra 17594  df-rgmod 17595  df-dsmm 18523  df-frlm 18538
This theorem is referenced by:  frlmelbas  18548  frlmfibas  18555  ellspd  18596  islindf4  18633  rrxbase  21548  rrxds  21553
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