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Theorem frlmlbs 26572
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 2358 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
41, 2, 3uvcff 26563 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> ( Base `  F ) )
5 frn 5478 . . 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 5115 . . . . . 6  |-  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  dom  a
8 eqid 2358 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
92, 8, 3frlmbasf 26551 . . . . . . . . 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 5472 . . . . . . . 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 5425 . . . . . . 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 3302 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  ( `' a " ( _V  \  { ( 0g `  R ) } ) )  C_  I )
1615ralrimiva 2702 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  A. a  e.  ( Base `  F
) ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I )
17 rabid2 2793 . . . 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 3273 . . . 4  |-  I  C_  I
20 eqid 2358 . . . . 5  |-  ( LSpan `  F )  =  (
LSpan `  F )
21 eqid 2358 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
22 eqid 2358 . . . . 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 26571 . . . 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 5472 . . . . . 6  |-  ( U : I --> ( Base `  F )  ->  U  Fn  I )
264, 25syl 15 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U  Fn  I )
27 fnima 5444 . . . . 5  |-  ( U  Fn  I  ->  ( U " I )  =  ran  U )
2826, 27syl 15 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( U " I )  =  ran  U )
2928fveq2d 5612 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  ( ( LSpan `  F ) `  ran  U ) )
3018, 24, 293eqtr2rd 2397 . 2  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ran  U )  =  ( Base `  F
) )
31 eqid 2358 . . . . . 6  |-  ( .s
`  F )  =  ( .s `  F
)
32 eqid 2358 . . . . . 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 3379 . . . . . . 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 2867 . . . . . . . 8  |-  c  e. 
_V
3837snid 3743 . . . . . . 7  |-  c  e. 
{ c }
39 snssi 3838 . . . . . . . . 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 3479 . . . . . . . 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 2445 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  c  e.  ( I  \  ( I 
\  { c } ) ) )
442frlmsca 26544 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
4544fveq2d 5612 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( Base `  R )  =  ( Base `  (Scalar `  F ) ) )
4644fveq2d 5612 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( 0g `  R )  =  ( 0g `  (Scalar `  F ) ) )
4746sneqd 3729 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  { ( 0g `  R ) }  =  { ( 0g `  (Scalar `  F ) ) } )
4845, 47difeq12d 3371 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( Base `  R )  \  { ( 0g `  R ) } )  =  ( ( Base `  (Scalar `  F )
)  \  { ( 0g `  (Scalar `  F
) ) } ) )
4948eleq2d 2425 . . . . . . . 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 26570 . . . . 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 16116 . . . . . . . . . . . . 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 26564 . . . . . . . . . . . 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 5342 . . . . . . . . . . . 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 5409 . . . . . . . . . 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 5521 . . . . . . . . . . . 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 5665 . . . . . . . . . . . 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 2363 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U " { c } )  =  { ( U `
 c ) } )
6964, 68difeq12d 3371 . . . . . . . . 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 2391 . . . . . . . 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 5612 . . . . . . 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 26571 . . . . . . . 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 2390 . . . . . 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 2425 . . . . 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 2711 . . 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 5953 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
b ( .s `  F ) a )  =  ( b ( .s `  F ) ( U `  c
) ) )
79 sneq 3727 . . . . . . . . . 10  |-  ( a  =  ( U `  c )  ->  { a }  =  { ( U `  c ) } )
8079difeq2d 3370 . . . . . . . . 9  |-  ( a  =  ( U `  c )  ->  ( ran  U  \  { a } )  =  ( ran  U  \  {
( U `  c
) } ) )
8180fveq2d 5612 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
( LSpan `  F ) `  ( ran  U  \  { a } ) )  =  ( (
LSpan `  F ) `  ( ran  U  \  {
( U `  c
) } ) ) )
8278, 81eleq12d 2426 . . . . . . 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 2639 . . . . 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 5751 . . . 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 5970 . . . 4  |-  ( R freeLMod  I )  e.  _V
892, 88eqeltri 2428 . . 3  |-  F  e. 
_V
90 eqid 2358 . . . 4  |-  (Scalar `  F )  =  (Scalar `  F )
91 eqid 2358 . . . 4  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
92 frlmlbs.j . . . 4  |-  J  =  (LBasis `  F )
93 eqid 2358 . . . 4  |-  ( 0g
`  (Scalar `  F )
)  =  ( 0g
`  (Scalar `  F )
)
943, 90, 31, 91, 92, 20, 93islbs 15928 . . 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 1642    e. wcel 1710   A.wral 2619   {crab 2623   _Vcvv 2864    \ cdif 3225    C_ wss 3228   {csn 3716   `'ccnv 4770   dom cdm 4771   ran crn 4772   "cima 4774   Fun wfun 5331    Fn wfn 5332   -->wf 5333   -1-1->wf1 5334   ` cfv 5337  (class class class)co 5945   Basecbs 13245  Scalarcsca 13308   .scvsca 13309   0gc0g 13499   Ringcrg 15436   LSpanclspn 15827  LBasisclbs 15926  NzRingcnzr 16108   freeLMod cfrlm 26535   unitVec cuvc 26536
This theorem is referenced by:  frlmup3  26575  frlmup4  26576  lmisfree  26635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-hom 13329  df-cco 13330  df-prds 13447  df-pws 13449  df-0g 13503  df-gsum 13504  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-ghm 14780  df-cntz 14892  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-subrg 15642  df-lmod 15728  df-lss 15789  df-lsp 15828  df-lmhm 15878  df-lbs 15927  df-sra 16024  df-rgmod 16025  df-nzr 16109  df-dsmm 26521  df-frlm 26537  df-uvc 26538
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