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Theorem lvecdim 15926
Description: The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 15913 and lbsacsbs 15925 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 14302. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
lvecdim.1  |-  J  =  (LBasis `  W )
Assertion
Ref Expression
lvecdim  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  ~~  T )

Proof of Theorem lvecdim
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
2 eqid 2296 . . . . 5  |-  (mrCls `  ( LSubSp `  W )
)  =  (mrCls `  ( LSubSp `  W )
)
3 eqid 2296 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
41, 2, 3lssacsex 15913 . . . 4  |-  ( W  e.  LVec  ->  ( (
LSubSp `  W )  e.  (ACS `  ( Base `  W ) )  /\  A. x  e.  ~P  ( Base `  W ) A. y  e.  ( Base `  W ) A. z  e.  ( ( (mrCls `  ( LSubSp `  W )
) `  ( x  u.  { y } ) )  \  ( (mrCls `  ( LSubSp `  W )
) `  x )
) y  e.  ( (mrCls `  ( LSubSp `  W ) ) `  ( x  u.  { z } ) ) ) )
543ad2ant1 976 . . 3  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  (
( LSubSp `  W )  e.  (ACS `  ( Base `  W ) )  /\  A. x  e.  ~P  ( Base `  W ) A. y  e.  ( Base `  W ) A. z  e.  ( ( (mrCls `  ( LSubSp `  W )
) `  ( x  u.  { y } ) )  \  ( (mrCls `  ( LSubSp `  W )
) `  x )
) y  e.  ( (mrCls `  ( LSubSp `  W ) ) `  ( x  u.  { z } ) ) ) )
65simpld 445 . 2  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  ( LSubSp `
 W )  e.  (ACS `  ( Base `  W ) ) )
7 eqid 2296 . 2  |-  (mrInd `  ( LSubSp `  W )
)  =  (mrInd `  ( LSubSp `  W )
)
85simprd 449 . 2  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  A. x  e.  ~P  ( Base `  W
) A. y  e.  ( Base `  W
) A. z  e.  ( ( (mrCls `  ( LSubSp `  W )
) `  ( x  u.  { y } ) )  \  ( (mrCls `  ( LSubSp `  W )
) `  x )
) y  e.  ( (mrCls `  ( LSubSp `  W ) ) `  ( x  u.  { z } ) ) )
9 simp2 956 . . . 4  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  e.  J )
10 lvecdim.1 . . . . . 6  |-  J  =  (LBasis `  W )
111, 2, 3, 7, 10lbsacsbs 15925 . . . . 5  |-  ( W  e.  LVec  ->  ( S  e.  J  <->  ( S  e.  (mrInd `  ( LSubSp `  W ) )  /\  ( (mrCls `  ( LSubSp `  W ) ) `  S )  =  (
Base `  W )
) ) )
12113ad2ant1 976 . . . 4  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  ( S  e.  J  <->  ( S  e.  (mrInd `  ( LSubSp `  W ) )  /\  ( (mrCls `  ( LSubSp `  W ) ) `  S )  =  (
Base `  W )
) ) )
139, 12mpbid 201 . . 3  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  ( S  e.  (mrInd `  ( LSubSp `
 W ) )  /\  ( (mrCls `  ( LSubSp `  W )
) `  S )  =  ( Base `  W
) ) )
1413simpld 445 . 2  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  e.  (mrInd `  ( LSubSp `  W ) ) )
15 simp3 957 . . . 4  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  T  e.  J )
161, 2, 3, 7, 10lbsacsbs 15925 . . . . 5  |-  ( W  e.  LVec  ->  ( T  e.  J  <->  ( T  e.  (mrInd `  ( LSubSp `  W ) )  /\  ( (mrCls `  ( LSubSp `  W ) ) `  T )  =  (
Base `  W )
) ) )
17163ad2ant1 976 . . . 4  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  ( T  e.  J  <->  ( T  e.  (mrInd `  ( LSubSp `  W ) )  /\  ( (mrCls `  ( LSubSp `  W ) ) `  T )  =  (
Base `  W )
) ) )
1815, 17mpbid 201 . . 3  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  ( T  e.  (mrInd `  ( LSubSp `
 W ) )  /\  ( (mrCls `  ( LSubSp `  W )
) `  T )  =  ( Base `  W
) ) )
1918simpld 445 . 2  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  T  e.  (mrInd `  ( LSubSp `  W ) ) )
2013simprd 449 . . 3  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  (
(mrCls `  ( LSubSp `  W ) ) `  S )  =  (
Base `  W )
)
2118simprd 449 . . 3  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  (
(mrCls `  ( LSubSp `  W ) ) `  T )  =  (
Base `  W )
)
2220, 21eqtr4d 2331 . 2  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  (
(mrCls `  ( LSubSp `  W ) ) `  S )  =  ( (mrCls `  ( LSubSp `  W ) ) `  T ) )
236, 2, 7, 8, 14, 19, 22acsexdimd 14302 1  |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  ~~  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    u. cun 3163   ~Pcpw 3638   {csn 3653   class class class wbr 4039   ` cfv 5271    ~~ cen 6876   Basecbs 13164  mrClscmrc 13501  mrIndcmri 13502  ACScacs 13503   LSubSpclss 15705  LBasisclbs 15843   LVecclvec 15871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358  ax-ac2 8105  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-r1 7452  df-rank 7453  df-card 7588  df-acn 7591  df-ac 7759  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-tset 13243  df-ple 13244  df-ocomp 13245  df-0g 13420  df-mre 13504  df-mrc 13505  df-mri 13506  df-acs 13507  df-preset 14078  df-drs 14079  df-poset 14096  df-ipo 14271  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lbs 15844  df-lvec 15872
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