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Theorem dia1dim2 31321
Description: Two expressions for a 1-dimensional subspace of partial vector space A (when  F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dia1dim2.h  |-  H  =  ( LHyp `  K
)
dia1dim2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia1dim2.r  |-  R  =  ( ( trL `  K
) `  W )
dva1dim2.u  |-  U  =  ( ( DVecA `  K
) `  W )
dia1dim2.i  |-  I  =  ( ( DIsoA `  K
) `  W )
dva1dim2.n  |-  N  =  ( LSpan `  U )
Assertion
Ref Expression
dia1dim2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( N `  { F } ) )

Proof of Theorem dia1dim2
Dummy variables  g 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dia1dim2.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
2 eqid 2358 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
3 dva1dim2.u . . . . . . 7  |-  U  =  ( ( DVecA `  K
) `  W )
4 eqid 2358 . . . . . . 7  |-  (Scalar `  U )  =  (Scalar `  U )
5 eqid 2358 . . . . . . 7  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
61, 2, 3, 4, 5dvabase 31265 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
76adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( Base `  (Scalar `  U )
)  =  ( (
TEndo `  K ) `  W ) )
87rexeqdv 2819 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. s  e.  ( Base `  (Scalar `  U )
) g  =  ( s ( .s `  U ) F )  <->  E. s  e.  (
( TEndo `  K ) `  W ) g  =  ( s ( .s
`  U ) F ) ) )
9 dia1dim2.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
10 eqid 2358 . . . . . . . 8  |-  ( .s
`  U )  =  ( .s `  U
)
111, 9, 2, 3, 10dvavsca 31275 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  ( ( TEndo `  K
) `  W )  /\  F  e.  T
) )  ->  (
s ( .s `  U ) F )  =  ( s `  F ) )
1211anass1rs 782 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  ( ( TEndo `  K
) `  W )
)  ->  ( s
( .s `  U
) F )  =  ( s `  F
) )
1312eqeq2d 2369 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  ( ( TEndo `  K
) `  W )
)  ->  ( g  =  ( s ( .s `  U ) F )  <->  g  =  ( s `  F
) ) )
1413rexbidva 2636 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. s  e.  ( ( TEndo `  K ) `  W ) g  =  ( s ( .s
`  U ) F )  <->  E. s  e.  ( ( TEndo `  K ) `  W ) g  =  ( s `  F
) ) )
158, 14bitrd 244 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. s  e.  ( Base `  (Scalar `  U )
) g  =  ( s ( .s `  U ) F )  <->  E. s  e.  (
( TEndo `  K ) `  W ) g  =  ( s `  F
) ) )
1615abbidv 2472 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { g  |  E. s  e.  (
Base `  (Scalar `  U
) ) g  =  ( s ( .s
`  U ) F ) }  =  {
g  |  E. s  e.  ( ( TEndo `  K
) `  W )
g  =  ( s `
 F ) } )
171, 3dvalvec 31285 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
1817adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  U  e.  LVec )
19 lveclmod 15958 . . . 4  |-  ( U  e.  LVec  ->  U  e. 
LMod )
2018, 19syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  U  e.  LMod )
21 simpr 447 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F  e.  T )
22 eqid 2358 . . . . . 6  |-  ( Base `  U )  =  (
Base `  U )
231, 9, 3, 22dvavbase 31271 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  T )
2423adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( Base `  U )  =  T )
2521, 24eleqtrrd 2435 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F  e.  ( Base `  U )
)
26 dva1dim2.n . . . 4  |-  N  =  ( LSpan `  U )
274, 5, 22, 10, 26lspsn 15858 . . 3  |-  ( ( U  e.  LMod  /\  F  e.  ( Base `  U
) )  ->  ( N `  { F } )  =  {
g  |  E. s  e.  ( Base `  (Scalar `  U ) ) g  =  ( s ( .s `  U ) F ) } )
2820, 25, 27syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( N `  { F } )  =  { g  |  E. s  e.  (
Base `  (Scalar `  U
) ) g  =  ( s ( .s
`  U ) F ) } )
29 dia1dim2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
30 dia1dim2.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
311, 9, 29, 2, 30dia1dim 31320 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
g  |  E. s  e.  ( ( TEndo `  K
) `  W )
g  =  ( s `
 F ) } )
3216, 28, 313eqtr4rd 2401 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( N `  { F } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   E.wrex 2620   {csn 3716   ` cfv 5337  (class class class)co 5945   Basecbs 13245  Scalarcsca 13308   .scvsca 13309   LModclmod 15726   LSpanclspn 15827   LVecclvec 15954   HLchlt 29609   LHypclh 30242   LTrncltrn 30359   trLctrl 30416   TEndoctendo 31010   DVecAcdveca 31260   DIsoAcdia 31287
This theorem is referenced by:  dia1dimid  31322  dia2dimlem5  31327  dia2dimlem10  31332
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-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-fal 1320  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-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-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-tpos 6321  df-undef 6385  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  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-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  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-0g 13503  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-dvr 15564  df-drng 15613  df-lmod 15728  df-lss 15789  df-lsp 15828  df-lvec 15955  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-llines 29756  df-lplanes 29757  df-lvols 29758  df-lines 29759  df-psubsp 29761  df-pmap 29762  df-padd 30054  df-lhyp 30246  df-laut 30247  df-ldil 30362  df-ltrn 30363  df-trl 30417  df-tgrp 31001  df-tendo 31013  df-edring 31015  df-dveca 31261  df-disoa 31288
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