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Theorem frlmlbs 27264
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 2442 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
41, 2, 3uvcff 27255 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> ( Base `  F ) )
5 frn 5626 . . 3  |-  ( U : I --> ( Base `  F )  ->  ran  U 
C_  ( Base `  F
) )
64, 5syl 16 . 2  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ran  U 
C_  ( Base `  F
) )
7 cnvimass 5253 . . . . . 6  |-  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  dom  a
8 eqid 2442 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
92, 8, 3frlmbasf 27243 . . . . . . . 8  |-  ( ( I  e.  V  /\  a  e.  ( Base `  F ) )  -> 
a : I --> ( Base `  R ) )
109adantll 696 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  a :
I --> ( Base `  R
) )
11 ffn 5620 . . . . . . 7  |-  ( a : I --> ( Base `  R )  ->  a  Fn  I )
12 fndm 5573 . . . . . . 7  |-  ( a  Fn  I  ->  dom  a  =  I )
1310, 11, 123syl 19 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  dom  a  =  I )
147, 13syl5sseq 3382 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  ( `' a " ( _V  \  { ( 0g `  R ) } ) )  C_  I )
1514ralrimiva 2795 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  A. a  e.  ( Base `  F
) ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I )
16 rabid2 2891 . . . 4  |-  ( (
Base `  F )  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I }  <->  A. a  e.  ( Base `  F ) ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  I
)
1715, 16sylibr 205 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( Base `  F )  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I }
)
18 ssid 3353 . . . 4  |-  I  C_  I
19 eqid 2442 . . . . 5  |-  ( LSpan `  F )  =  (
LSpan `  F )
20 eqid 2442 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
21 eqid 2442 . . . . 5  |-  { a  e.  ( Base `  F
)  |  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  I }  =  { a  e.  ( Base `  F
)  |  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  I }
222, 1, 19, 3, 20, 21frlmsslsp 27263 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V  /\  I  C_  I )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  {
a  e.  ( Base `  F )  |  ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  C_  I } )
2318, 22mp3an3 1269 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  {
a  e.  ( Base `  F )  |  ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  C_  I } )
24 ffn 5620 . . . . 5  |-  ( U : I --> ( Base `  F )  ->  U  Fn  I )
25 fnima 5592 . . . . 5  |-  ( U  Fn  I  ->  ( U " I )  =  ran  U )
264, 24, 253syl 19 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( U " I )  =  ran  U )
2726fveq2d 5761 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  ( ( LSpan `  F ) `  ran  U ) )
2817, 23, 273eqtr2rd 2481 . 2  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ran  U )  =  ( Base `  F
) )
29 eqid 2442 . . . . . 6  |-  ( .s
`  F )  =  ( .s `  F
)
30 eqid 2442 . . . . . 6  |-  { a  e.  ( Base `  F
)  |  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  (
I  \  { c } ) }  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( I  \  { c } ) }
31 simpll 732 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  R  e.  Ring )
32 simplr 733 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  I  e.  V
)
33 difssd 3461 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( I  \  { c } ) 
C_  I )
34 vex 2965 . . . . . . . 8  |-  c  e. 
_V
3534snid 3865 . . . . . . 7  |-  c  e. 
{ c }
36 snssi 3966 . . . . . . . . 9  |-  ( c  e.  I  ->  { c }  C_  I )
3736ad2antrl 710 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  { c } 
C_  I )
38 dfss4 3560 . . . . . . . 8  |-  ( { c }  C_  I  <->  ( I  \  ( I 
\  { c } ) )  =  {
c } )
3937, 38sylib 190 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( I  \ 
( I  \  {
c } ) )  =  { c } )
4035, 39syl5eleqr 2529 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  c  e.  ( I  \  ( I 
\  { c } ) ) )
412frlmsca 27236 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
4241fveq2d 5761 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( Base `  R )  =  ( Base `  (Scalar `  F ) ) )
4341fveq2d 5761 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( 0g `  R )  =  ( 0g `  (Scalar `  F ) ) )
4443sneqd 3851 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  { ( 0g `  R ) }  =  { ( 0g `  (Scalar `  F ) ) } )
4542, 44difeq12d 3452 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( Base `  R )  \  { ( 0g `  R ) } )  =  ( ( Base `  (Scalar `  F )
)  \  { ( 0g `  (Scalar `  F
) ) } ) )
4645eleq2d 2509 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
b  e.  ( (
Base `  R )  \  { ( 0g `  R ) } )  <-> 
b  e.  ( (
Base `  (Scalar `  F
) )  \  {
( 0g `  (Scalar `  F ) ) } ) ) )
4746biimpar 473 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) )  -> 
b  e.  ( (
Base `  R )  \  { ( 0g `  R ) } ) )
4847adantrl 698 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  b  e.  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )
492, 1, 3, 8, 29, 20, 30, 31, 32, 33, 40, 48frlmssuvc2 27262 . . . . 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 } ) } )
5020, 8rngelnzr 16367 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  b  e.  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  ->  R  e. NzRing )
5131, 48, 50syl2anc 644 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  R  e. NzRing )
521, 2, 3uvcf1 27256 . . . . . . . . . 10  |-  ( ( R  e. NzRing  /\  I  e.  V )  ->  U : I -1-1-> ( Base `  F ) )
5351, 32, 52syl2anc 644 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  U : I
-1-1-> ( Base `  F
) )
54 df-f1 5488 . . . . . . . . . 10  |-  ( U : I -1-1-> ( Base `  F )  <->  ( U : I --> ( Base `  F )  /\  Fun  `' U ) )
5554simprbi 452 . . . . . . . . 9  |-  ( U : I -1-1-> ( Base `  F )  ->  Fun  `' U )
56 imadif 5557 . . . . . . . . 9  |-  ( Fun  `' U  ->  ( U
" ( I  \  { c } ) )  =  ( ( U " I ) 
\  ( U " { c } ) ) )
5753, 55, 563syl 19 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U "
( I  \  {
c } ) )  =  ( ( U
" I )  \ 
( U " {
c } ) ) )
58 f1fn 5669 . . . . . . . . . 10  |-  ( U : I -1-1-> ( Base `  F )  ->  U  Fn  I )
5953, 58, 253syl 19 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U "
I )  =  ran  U )
6053, 58syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  U  Fn  I
)
61 simprl 734 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  c  e.  I
)
62 fnsnfv 5815 . . . . . . . . . . 11  |-  ( ( U  Fn  I  /\  c  e.  I )  ->  { ( U `  c ) }  =  ( U " { c } ) )
6360, 61, 62syl2anc 644 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  { ( U `
 c ) }  =  ( U " { c } ) )
6463eqcomd 2447 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U " { c } )  =  { ( U `
 c ) } )
6559, 64difeq12d 3452 . . . . . . . 8  |-  ( ( ( 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 ) } ) )
6657, 65eqtr2d 2475 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( ran  U  \  { ( U `  c ) } )  =  ( U "
( I  \  {
c } ) ) )
6766fveq2d 5761 . . . . . 6  |-  ( ( ( 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 } ) ) ) )
682, 1, 19, 3, 20, 30frlmsslsp 27263 . . . . . . 7  |-  ( ( 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 } ) } )
6931, 32, 33, 68syl3anc 1185 . . . . . 6  |-  ( ( ( 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 } ) } )
7067, 69eqtrd 2474 . . . . 5  |-  ( ( ( 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 } ) } )
7149, 70neleqtrrd 2538 . . . 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 ) } ) ) )
7271ralrimivva 2804 . . 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 ) } ) ) )
73 oveq2 6118 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
b ( .s `  F ) a )  =  ( b ( .s `  F ) ( U `  c
) ) )
74 sneq 3849 . . . . . . . . . 10  |-  ( a  =  ( U `  c )  ->  { a }  =  { ( U `  c ) } )
7574difeq2d 3451 . . . . . . . . 9  |-  ( a  =  ( U `  c )  ->  ( ran  U  \  { a } )  =  ( ran  U  \  {
( U `  c
) } ) )
7675fveq2d 5761 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
( LSpan `  F ) `  ( ran  U  \  { a } ) )  =  ( (
LSpan `  F ) `  ( ran  U  \  {
( U `  c
) } ) ) )
7773, 76eleq12d 2510 . . . . . . 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
) } ) ) ) )
7877notbid 287 . . . . . 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
) } ) ) ) )
7978ralbidv 2731 . . . . 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 ) } ) ) ) )
8079ralrn 5902 . . . 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 ) } ) ) ) )
814, 24, 803syl 19 . . 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 ) } ) ) ) )
8272, 81mpbird 225 . 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 } ) ) )
83 ovex 6135 . . . 4  |-  ( R freeLMod  I )  e.  _V
842, 83eqeltri 2512 . . 3  |-  F  e. 
_V
85 eqid 2442 . . . 4  |-  (Scalar `  F )  =  (Scalar `  F )
86 eqid 2442 . . . 4  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
87 frlmlbs.j . . . 4  |-  J  =  (LBasis `  F )
88 eqid 2442 . . . 4  |-  ( 0g
`  (Scalar `  F )
)  =  ( 0g
`  (Scalar `  F )
)
893, 85, 29, 86, 87, 19, 88islbs 16179 . . 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 } ) ) ) ) )
9084, 89ax-mp 5 . 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 } ) ) ) )
916, 28, 82, 90syl3anbrc 1139 1  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ran  U  e.  J )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   A.wral 2711   {crab 2715   _Vcvv 2962    \ cdif 3303    C_ wss 3306   {csn 3838   `'ccnv 4906   dom cdm 4907   ran crn 4908   "cima 4910   Fun wfun 5477    Fn wfn 5478   -->wf 5479   -1-1->wf1 5480   ` cfv 5483  (class class class)co 6110   Basecbs 13500  Scalarcsca 13563   .scvsca 13564   0gc0g 13754   Ringcrg 15691   LSpanclspn 16078  LBasisclbs 16177  NzRingcnzr 16359   freeLMod cfrlm 27227   unitVec cuvc 27228
This theorem is referenced by:  frlmup3  27267  frlmup4  27268  lmisfree  27327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-sup 7475  df-oi 7508  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-fz 11075  df-fzo 11167  df-seq 11355  df-hash 11650  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-hom 13584  df-cco 13585  df-prds 13702  df-pws 13704  df-0g 13758  df-gsum 13759  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-mhm 14769  df-submnd 14770  df-grp 14843  df-minusg 14844  df-sbg 14845  df-mulg 14846  df-subg 14972  df-ghm 15035  df-cntz 15147  df-cmn 15445  df-abl 15446  df-mgp 15680  df-rng 15694  df-ur 15696  df-subrg 15897  df-lmod 15983  df-lss 16040  df-lsp 16079  df-lmhm 16129  df-lbs 16178  df-sra 16275  df-rgmod 16276  df-nzr 16360  df-dsmm 27213  df-frlm 27229  df-uvc 27230
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