Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frlmlbs Unicode version

Theorem frlmlbs 27121
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 2408 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
41, 2, 3uvcff 27112 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> ( Base `  F ) )
5 frn 5560 . . 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 5187 . . . . . 6  |-  ( `' a " ( _V 
\  { ( 0g
`  R ) } ) )  C_  dom  a
8 eqid 2408 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
92, 8, 3frlmbasf 27100 . . . . . . . 8  |-  ( ( I  e.  V  /\  a  e.  ( Base `  F ) )  -> 
a : I --> ( Base `  R ) )
109adantll 695 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  a :
I --> ( Base `  R
) )
11 ffn 5554 . . . . . . 7  |-  ( a : I --> ( Base `  R )  ->  a  Fn  I )
12 fndm 5507 . . . . . . 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 3360 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  a  e.  (
Base `  F )
)  ->  ( `' a " ( _V  \  { ( 0g `  R ) } ) )  C_  I )
1514ralrimiva 2753 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  A. a  e.  ( Base `  F
) ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I )
16 rabid2 2849 . . . 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 204 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( Base `  F )  =  { a  e.  (
Base `  F )  |  ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  C_  I }
)
18 ssid 3331 . . . 4  |-  I  C_  I
19 eqid 2408 . . . . 5  |-  ( LSpan `  F )  =  (
LSpan `  F )
20 eqid 2408 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
21 eqid 2408 . . . . 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 27120 . . . 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 1268 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  {
a  e.  ( Base `  F )  |  ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  C_  I } )
24 ffn 5554 . . . . 5  |-  ( U : I --> ( Base `  F )  ->  U  Fn  I )
25 fnima 5526 . . . . 5  |-  ( U  Fn  I  ->  ( U " I )  =  ran  U )
264, 24, 253syl 19 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( U " I )  =  ran  U )
2726fveq2d 5695 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ( U " I
) )  =  ( ( LSpan `  F ) `  ran  U ) )
2817, 23, 273eqtr2rd 2447 . 2  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( LSpan `  F ) `  ran  U )  =  ( Base `  F
) )
29 eqid 2408 . . . . . 6  |-  ( .s
`  F )  =  ( .s `  F
)
30 eqid 2408 . . . . . 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 731 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  R  e.  Ring )
32 simplr 732 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  I  e.  V
)
33 difssd 3439 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( I  \  { c } ) 
C_  I )
34 vex 2923 . . . . . . . 8  |-  c  e. 
_V
3534snid 3805 . . . . . . 7  |-  c  e. 
{ c }
36 snssi 3906 . . . . . . . . 9  |-  ( c  e.  I  ->  { c }  C_  I )
3736ad2antrl 709 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  { c } 
C_  I )
38 dfss4 3539 . . . . . . . 8  |-  ( { c }  C_  I  <->  ( I  \  ( I 
\  { c } ) )  =  {
c } )
3937, 38sylib 189 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( I  \ 
( I  \  {
c } ) )  =  { c } )
4035, 39syl5eleqr 2495 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  c  e.  ( I  \  ( I 
\  { c } ) ) )
412frlmsca 27093 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
4241fveq2d 5695 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( Base `  R )  =  ( Base `  (Scalar `  F ) ) )
4341fveq2d 5695 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ( 0g `  R )  =  ( 0g `  (Scalar `  F ) ) )
4443sneqd 3791 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  { ( 0g `  R ) }  =  { ( 0g `  (Scalar `  F ) ) } )
4542, 44difeq12d 3430 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
( Base `  R )  \  { ( 0g `  R ) } )  =  ( ( Base `  (Scalar `  F )
)  \  { ( 0g `  (Scalar `  F
) ) } ) )
4645eleq2d 2475 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  (
b  e.  ( (
Base `  R )  \  { ( 0g `  R ) } )  <-> 
b  e.  ( (
Base `  (Scalar `  F
) )  \  {
( 0g `  (Scalar `  F ) ) } ) ) )
4746biimpar 472 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) )  -> 
b  e.  ( (
Base `  R )  \  { ( 0g `  R ) } ) )
4847adantrl 697 . . . . . 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 27119 . . . . 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 16295 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  b  e.  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  ->  R  e. NzRing )
5131, 48, 50syl2anc 643 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  R  e. NzRing )
521, 2, 3uvcf1 27113 . . . . . . . . . 10  |-  ( ( R  e. NzRing  /\  I  e.  V )  ->  U : I -1-1-> ( Base `  F ) )
5351, 32, 52syl2anc 643 . . . . . . . . 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 5422 . . . . . . . . . 10  |-  ( U : I -1-1-> ( Base `  F )  <->  ( U : I --> ( Base `  F )  /\  Fun  `' U ) )
5554simprbi 451 . . . . . . . . 9  |-  ( U : I -1-1-> ( Base `  F )  ->  Fun  `' U )
56 imadif 5491 . . . . . . . . 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 5603 . . . . . . . . . 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 733 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  c  e.  I
)
62 fnsnfv 5749 . . . . . . . . . . 11  |-  ( ( U  Fn  I  /\  c  e.  I )  ->  { ( U `  c ) }  =  ( U " { c } ) )
6360, 61, 62syl2anc 643 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  { ( U `
 c ) }  =  ( U " { c } ) )
6463eqcomd 2413 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  I  e.  V )  /\  ( c  e.  I  /\  b  e.  ( ( Base `  (Scalar `  F ) )  \  { ( 0g `  (Scalar `  F ) ) } ) ) )  ->  ( U " { c } )  =  { ( U `
 c ) } )
6559, 64difeq12d 3430 . . . . . . . 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 2441 . . . . . . 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 5695 . . . . . 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 27120 . . . . . . 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 1184 . . . . . 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 2440 . . . . 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 2504 . . . 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 2762 . . 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 6052 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
b ( .s `  F ) a )  =  ( b ( .s `  F ) ( U `  c
) ) )
74 sneq 3789 . . . . . . . . . 10  |-  ( a  =  ( U `  c )  ->  { a }  =  { ( U `  c ) } )
7574difeq2d 3429 . . . . . . . . 9  |-  ( a  =  ( U `  c )  ->  ( ran  U  \  { a } )  =  ( ran  U  \  {
( U `  c
) } ) )
7675fveq2d 5695 . . . . . . . 8  |-  ( a  =  ( U `  c )  ->  (
( LSpan `  F ) `  ( ran  U  \  { a } ) )  =  ( (
LSpan `  F ) `  ( ran  U  \  {
( U `  c
) } ) ) )
7773, 76eleq12d 2476 . . . . . . 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 286 . . . . . 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 2690 . . . . 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 5836 . . . 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 224 . 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 6069 . . . 4  |-  ( R freeLMod  I )  e.  _V
842, 83eqeltri 2478 . . 3  |-  F  e. 
_V
85 eqid 2408 . . . 4  |-  (Scalar `  F )  =  (Scalar `  F )
86 eqid 2408 . . . 4  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
87 frlmlbs.j . . . 4  |-  J  =  (LBasis `  F )
88 eqid 2408 . . . 4  |-  ( 0g
`  (Scalar `  F )
)  =  ( 0g
`  (Scalar `  F )
)
893, 85, 29, 86, 87, 19, 88islbs 16107 . . 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 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 } ) ) ) )
916, 28, 82, 90syl3anbrc 1138 1  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  ran  U  e.  J )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   {crab 2674   _Vcvv 2920    \ cdif 3281    C_ wss 3284   {csn 3778   `'ccnv 4840   dom cdm 4841   ran crn 4842   "cima 4844   Fun wfun 5411    Fn wfn 5412   -->wf 5413   -1-1->wf1 5414   ` cfv 5417  (class class class)co 6044   Basecbs 13428  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   Ringcrg 15619   LSpanclspn 16006  LBasisclbs 16105  NzRingcnzr 16287   freeLMod cfrlm 27084   unitVec cuvc 27085
This theorem is referenced by:  frlmup3  27124  frlmup4  27125  lmisfree  27184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-fz 11004  df-fzo 11095  df-seq 11283  df-hash 11578  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-tset 13507  df-ple 13508  df-ds 13510  df-hom 13512  df-cco 13513  df-prds 13630  df-pws 13632  df-0g 13686  df-gsum 13687  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-mhm 14697  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-mulg 14774  df-subg 14900  df-ghm 14963  df-cntz 15075  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-subrg 15825  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lmhm 16057  df-lbs 16106  df-sra 16203  df-rgmod 16204  df-nzr 16288  df-dsmm 27070  df-frlm 27086  df-uvc 27087
  Copyright terms: Public domain W3C validator