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Theorem lvecdim 15910
Description: The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 15897 and lbsacsbs 15909 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 14286. (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 2283 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
2 eqid 2283 . . . . 5  |-  (mrCls `  ( LSubSp `  W )
)  =  (mrCls `  ( LSubSp `  W )
)
3 eqid 2283 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
41, 2, 3lssacsex 15897 . . . 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 2283 . 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 15909 . . . . 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 15909 . . . . 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 2318 . 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 14286 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 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    u. cun 3150   ~Pcpw 3625   {csn 3640   class class class wbr 4023   ` cfv 5255    ~~ cen 6860   Basecbs 13148  mrClscmrc 13485  mrIndcmri 13486  ACScacs 13487   LSubSpclss 15689  LBasisclbs 15827   LVecclvec 15855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342  ax-ac2 8089  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-r1 7436  df-rank 7437  df-card 7572  df-acn 7575  df-ac 7743  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-tset 13227  df-ple 13228  df-ocomp 13229  df-0g 13404  df-mre 13488  df-mrc 13489  df-mri 13490  df-acs 13491  df-preset 14062  df-drs 14063  df-poset 14080  df-ipo 14255  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lbs 15828  df-lvec 15856
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