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Theorem frlmlbs 27249
Description: The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.)
Hypotheses
Ref Expression
frlmlbs.f  |-  F  =  ( R freeLMod  I )
frlmlbs.u  |-  U  =  ( R unitVec  I )
frlmlbs.j  |-  J  =  (LBasis `  F )
Assertion
Ref Expression
frlmlbs  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ran  U  e.  J )

Proof of Theorem frlmlbs
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmlbs.u . . . 4  |-  U  =  ( R unitVec  I )
2 frlmlbs.f . . . 4  |-  F  =  ( R freeLMod  I )
3 eqid 2283 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
41, 2, 3uvcff 27240 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> ( Base `  F ) )
5 frn 5395 . . 3  |-  ( U : I --> ( Base `  F )  ->  ran  U 
C_  ( Base `  F
) )
64, 5syl 15 . 2  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ran  U 
C_  ( Base `  F
) )
7 cnvimass 5033 . . . . . 6  |-  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  dom  a
8 eqid 2283 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
92, 8, 3frlmbasf 27228 . . . . . . . . 9  |-  ( ( I  e.  V  /\  a  e.  ( Base `  F ) )  -> 
a : I --> ( Base `  R ) )
109adantll 694 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  a :
I --> ( Base `  R
) )
11 ffn 5389 . . . . . . . 8  |-  ( a : I --> ( Base `  R )  ->  a  Fn  I )
1210, 11syl 15 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  a  Fn  I )
13 fndm 5343 . . . . . . 7  |-  ( a  Fn  I  ->  dom  a  =  I )
1412, 13syl 15 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  dom  a  =  I )
157, 14syl5sseq 3226 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  ( `' a " ( _V  \  { ( 0g `  R ) } ) )  C_  I )
1615ralrimiva 2626 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  A. a  e.  ( Base `  F
) ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I )
17 rabid2 2717 . . . 4  |-  ( (
Base `  F )  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I }  <->  A. a  e.  ( Base `  F ) ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  I
)
1816, 17sylibr 203 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( Base `  F )  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I }
)
19 ssid 3197 . . . 4  |-  I  C_  I
20 eqid 2283 . . . . 5  |-  ( LSpan `  F )  =  (
LSpan `  F )
21 eqid 2283 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
22 eqid 2283 . . . . 5  |-  { a  e.  ( Base `  F
)  |  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  I }  =  { a  e.  ( Base `  F
)  |  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  I }
232, 1, 20, 3, 21, 22frlmsslsp 27248 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V  /\  I  C_  I )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  {
a  e.  ( Base `  F )  |  ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  C_  I } )
2419, 23mp3an3 1266 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  {
a  e.  ( Base `  F )  |  ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  C_  I } )
25 ffn 5389 . . . . . 6  |-  ( U : I --> ( Base `  F )  ->  U  Fn  I )
264, 25syl 15 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U  Fn  I )
27 fnima 5362 . . . . 5  |-  ( U  Fn  I  ->  ( U " I )  =  ran  U )
2826, 27syl 15 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( U " I )  =  ran  U )
2928fveq2d 5529 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  ( ( LSpan `  F ) `  ran  U ) )
3018, 24, 293eqtr2rd 2322 . 2  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ran  U )  =  ( Base `  F
) )
31 eqid 2283 . . . . . 6  |-  ( .s
`  F )  =  ( .s `  F
)
32 eqid 2283 . . . . . 6  |-  { a  e.  ( Base `  F
)  |  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  (
I  \  { c } ) }  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( I  \  { c } ) }
33 simpll 730 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  R  e.  Ring )
34 simplr 731 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  I  e.  V
)
35 difss 3303 . . . . . . 7  |-  ( I 
\  { c } )  C_  I
3635a1i 10 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( I  \  { c } ) 
C_  I )
37 vex 2791 . . . . . . . 8  |-  c  e. 
_V
3837snid 3667 . . . . . . 7  |-  c  e. 
{ c }
39 snssi 3759 . . . . . . . . 9  |-  ( c  e.  I  ->  { c }  C_  I )
4039ad2antrl 708 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  { c } 
C_  I )
41 dfss4 3403 . . . . . . . 8  |-  ( { c }  C_  I  <->  ( I  \  ( I 
\  { c } ) )  =  {
c } )
4240, 41sylib 188 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( I  \ 
( I  \  {
c } ) )  =  { c } )
4338, 42syl5eleqr 2370 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  c  e.  ( I  \  ( I 
\  { c } ) ) )
442frlmsca 27221 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
4544fveq2d 5529 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( Base `  R )  =  ( Base `  (Scalar `  F ) ) )
4644fveq2d 5529 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( 0g `  R )  =  ( 0g `  (Scalar `  F ) ) )
4746sneqd 3653 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  { ( 0g `  R ) }  =  { ( 0g `  (Scalar `  F ) ) } )
4845, 47difeq12d 3295 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( Base `  R )  \  { ( 0g `  R ) } )  =  ( ( Base `  (Scalar `  F )
)  \  { ( 0g `  (Scalar `  F
) ) } ) )
4948eleq2d 2350 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
b  e.  ( (
Base `  R )  \  { ( 0g `  R ) } )  <-> 
b  e.  ( (
Base `  (Scalar `  F
) )  \  {
( 0g `  (Scalar `  F ) ) } ) ) )
5049biimpar 471 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) )  -> 
b  e.  ( (
Base `  R )  \  { ( 0g `  R ) } ) )
5150adantrl 696 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  b  e.  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )
522, 1, 3, 8, 31, 21, 32, 33, 34, 36, 43, 51frlmssuvc2 27247 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  -.  ( b
( .s `  F
) ( U `  c ) )  e. 
{ a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( I  \  { c } ) } )
5321, 8rngelnzr 16017 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  b  e.  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  ->  R  e. NzRing )
5433, 51, 53syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  R  e. NzRing )
551, 2, 3uvcf1 27241 . . . . . . . . . . . 12  |-  ( ( R  e. NzRing  /\  I  e.  V )  ->  U : I -1-1-> ( Base `  F ) )
5654, 34, 55syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  U : I
-1-1-> ( Base `  F
) )
57 df-f1 5260 . . . . . . . . . . . 12  |-  ( U : I -1-1-> ( Base `  F )  <->  ( U : I --> ( Base `  F )  /\  Fun  `' U ) )
5857simprbi 450 . . . . . . . . . . 11  |-  ( U : I -1-1-> ( Base `  F )  ->  Fun  `' U )
5956, 58syl 15 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  Fun  `' U
)
60 imadif 5327 . . . . . . . . . 10  |-  ( Fun  `' U  ->  ( U
" ( I  \  { c } ) )  =  ( ( U " I ) 
\  ( U " { c } ) ) )
6159, 60syl 15 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U "
( I  \  {
c } ) )  =  ( ( U
" I )  \ 
( U " {
c } ) ) )
62 f1fn 5438 . . . . . . . . . . . 12  |-  ( U : I -1-1-> ( Base `  F )  ->  U  Fn  I )
6356, 62syl 15 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  U  Fn  I
)
6463, 27syl 15 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U "
I )  =  ran  U )
65 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  c  e.  I
)
66 fnsnfv 5582 . . . . . . . . . . . 12  |-  ( ( U  Fn  I  /\  c  e.  I )  ->  { ( U `  c ) }  =  ( U " { c } ) )
6763, 65, 66syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  { ( U `
 c ) }  =  ( U " { c } ) )
6867eqcomd 2288 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U " { c } )  =  { ( U `
 c ) } )
6964, 68difeq12d 3295 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( ( U
" I )  \ 
( U " {
c } ) )  =  ( ran  U  \  { ( U `  c ) } ) )
7061, 69eqtr2d 2316 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( ran  U  \  { ( U `  c ) } )  =  ( U "
( I  \  {
c } ) ) )
7170fveq2d 5529 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( ( LSpan `  F ) `  ( ran  U  \  { ( U `  c ) } ) )  =  ( ( LSpan `  F
) `  ( U " ( I  \  {
c } ) ) ) )
722, 1, 20, 3, 21, 32frlmsslsp 27248 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  I  e.  V  /\  (
I  \  { c } )  C_  I
)  ->  ( ( LSpan `  F ) `  ( U " ( I 
\  { c } ) ) )  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( I  \  { c } ) } )
7333, 34, 36, 72syl3anc 1182 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( ( LSpan `  F ) `  ( U " ( I  \  { c } ) ) )  =  {
a  e.  ( Base `  F )  |  ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  C_  (
I  \  { c } ) } )
7471, 73eqtrd 2315 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( ( LSpan `  F ) `  ( ran  U  \  { ( U `  c ) } ) )  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( I  \  { c } ) } )
7574eleq2d 2350 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( ( b ( .s `  F
) ( U `  c ) )  e.  ( ( LSpan `  F
) `  ( ran  U 
\  { ( U `
 c ) } ) )  <->  ( b
( .s `  F
) ( U `  c ) )  e. 
{ a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( I  \  { c } ) } ) )
7652, 75mtbird 292 . . . 4  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  -.  ( b
( .s `  F
) ( U `  c ) )  e.  ( ( LSpan `  F
) `  ( ran  U 
\  { ( U `
 c ) } ) ) )
7776ralrimivva 2635 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  A. c  e.  I  A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) ( U `
 c ) )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { ( U `  c ) } ) ) )
78 oveq2 5866 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
b ( .s `  F ) a )  =  ( b ( .s `  F ) ( U `  c
) ) )
79 sneq 3651 . . . . . . . . . 10  |-  ( a  =  ( U `  c )  ->  { a }  =  { ( U `  c ) } )
8079difeq2d 3294 . . . . . . . . 9  |-  ( a  =  ( U `  c )  ->  ( ran  U  \  { a } )  =  ( ran  U  \  {
( U `  c
) } ) )
8180fveq2d 5529 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
( LSpan `  F ) `  ( ran  U  \  { a } ) )  =  ( (
LSpan `  F ) `  ( ran  U  \  {
( U `  c
) } ) ) )
8278, 81eleq12d 2351 . . . . . . 7  |-  ( a  =  ( U `  c )  ->  (
( b ( .s
`  F ) a )  e.  ( (
LSpan `  F ) `  ( ran  U  \  {
a } ) )  <-> 
( b ( .s
`  F ) ( U `  c ) )  e.  ( (
LSpan `  F ) `  ( ran  U  \  {
( U `  c
) } ) ) ) )
8382notbid 285 . . . . . 6  |-  ( a  =  ( U `  c )  ->  ( -.  ( b ( .s
`  F ) a )  e.  ( (
LSpan `  F ) `  ( ran  U  \  {
a } ) )  <->  -.  ( b ( .s
`  F ) ( U `  c ) )  e.  ( (
LSpan `  F ) `  ( ran  U  \  {
( U `  c
) } ) ) ) )
8483ralbidv 2563 . . . . 5  |-  ( a  =  ( U `  c )  ->  ( A. b  e.  (
( Base `  (Scalar `  F
) )  \  {
( 0g `  (Scalar `  F ) ) } )  -.  ( b ( .s `  F
) a )  e.  ( ( LSpan `  F
) `  ( ran  U 
\  { a } ) )  <->  A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) ( U `
 c ) )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { ( U `  c ) } ) ) ) )
8584ralrn 5668 . . . 4  |-  ( U  Fn  I  ->  ( A. a  e.  ran  U A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) a )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { a } ) )  <->  A. c  e.  I  A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) ( U `
 c ) )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { ( U `  c ) } ) ) ) )
8626, 85syl 15 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( A. a  e.  ran  U A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) a )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { a } ) )  <->  A. c  e.  I  A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) ( U `
 c ) )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { ( U `  c ) } ) ) ) )
8777, 86mpbird 223 . 2  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  A. a  e.  ran  U A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) a )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { a } ) ) )
88 ovex 5883 . . . 4  |-  ( R freeLMod  I )  e.  _V
892, 88eqeltri 2353 . . 3  |-  F  e. 
_V
90 eqid 2283 . . . 4  |-  (Scalar `  F )  =  (Scalar `  F )
91 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
92 frlmlbs.j . . . 4  |-  J  =  (LBasis `  F )
93 eqid 2283 . . . 4  |-  ( 0g
`  (Scalar `  F )
)  =  ( 0g
`  (Scalar `  F )
)
943, 90, 31, 91, 92, 20, 93islbs 15829 . . 3  |-  ( F  e.  _V  ->  ( ran  U  e.  J  <->  ( ran  U 
C_  ( Base `  F
)  /\  ( ( LSpan `  F ) `  ran  U )  =  (
Base `  F )  /\  A. a  e.  ran  U A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) a )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { a } ) ) ) ) )
9589, 94ax-mp 8 . 2  |-  ( ran 
U  e.  J  <->  ( ran  U 
C_  ( Base `  F
)  /\  ( ( LSpan `  F ) `  ran  U )  =  (
Base `  F )  /\  A. a  e.  ran  U A. b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } )  -.  (
b ( .s `  F ) a )  e.  ( ( LSpan `  F ) `  ( ran  U  \  { a } ) ) ) )
966, 30, 87, 95syl3anbrc 1136 1  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ran  U  e.  J )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Ringcrg 15337   LSpanclspn 15728  LBasisclbs 15827  NzRingcnzr 16009   freeLMod cfrlm 27212   unitVec cuvc 27213
This theorem is referenced by:  frlmup3  27252  frlmup4  27253  lmisfree  27312
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-inf2 7342  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-rmo 2551  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-iin 3908  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  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-oi 7225  df-card 7572  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-fzo 10871  df-seq 11047  df-hash 11338  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-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lbs 15828  df-sra 15925  df-rgmod 15926  df-nzr 16010  df-dsmm 27198  df-frlm 27214  df-uvc 27215
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