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Theorem lsatcmp 29193
Description: If two atoms are comparable, they are equal. (atsseq 22927 analog.) TODO: can lspsncmp 15869 shorten this? (Contributed by NM, 25-Aug-2014.)
Hypotheses
Ref Expression
lsatcmp.a  |-  A  =  (LSAtoms `  W )
lsatcmp.w  |-  ( ph  ->  W  e.  LVec )
lsatcmp.t  |-  ( ph  ->  T  e.  A )
lsatcmp.u  |-  ( ph  ->  U  e.  A )
Assertion
Ref Expression
lsatcmp  |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )

Proof of Theorem lsatcmp
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lsatcmp.u . . 3  |-  ( ph  ->  U  e.  A )
2 lsatcmp.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
3 lveclmod 15859 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
42, 3syl 15 . . . 4  |-  ( ph  ->  W  e.  LMod )
5 eqid 2283 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
6 eqid 2283 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
7 eqid 2283 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
8 lsatcmp.a . . . . 5  |-  A  =  (LSAtoms `  W )
95, 6, 7, 8islsat 29181 . . . 4  |-  ( W  e.  LMod  ->  ( U  e.  A  <->  E. v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) U  =  ( ( LSpan `  W ) `  { v } ) ) )
104, 9syl 15 . . 3  |-  ( ph  ->  ( U  e.  A  <->  E. v  e.  ( (
Base `  W )  \  { ( 0g `  W ) } ) U  =  ( (
LSpan `  W ) `  { v } ) ) )
111, 10mpbid 201 . 2  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) U  =  ( ( LSpan `  W ) `  { v } ) )
12 eldifsn 3749 . . . . 5  |-  ( v  e.  ( ( Base `  W )  \  {
( 0g `  W
) } )  <->  ( v  e.  ( Base `  W
)  /\  v  =/=  ( 0g `  W ) ) )
13 lsatcmp.t . . . . . . . . . . 11  |-  ( ph  ->  T  e.  A )
147, 8, 4, 13lsatn0 29189 . . . . . . . . . 10  |-  ( ph  ->  T  =/=  { ( 0g `  W ) } )
1514ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  =/=  {
( 0g `  W
) } )
162ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  W  e.  LVec )
17 eqid 2283 . . . . . . . . . . . . . 14  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1817, 8, 4, 13lsatlssel 29187 . . . . . . . . . . . . 13  |-  ( ph  ->  T  e.  ( LSubSp `  W ) )
1918ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  e.  (
LSubSp `  W ) )
20 simplrl 736 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  v  e.  (
Base `  W )
)
21 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  C_  (
( LSpan `  W ) `  { v } ) )
225, 7, 17, 6lspsnat 15898 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LVec  /\  T  e.  ( LSubSp `  W )  /\  v  e.  ( Base `  W
) )  /\  T  C_  ( ( LSpan `  W
) `  { v } ) )  -> 
( T  =  ( ( LSpan `  W ) `  { v } )  \/  T  =  {
( 0g `  W
) } ) )
2316, 19, 20, 21, 22syl31anc 1185 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  ( T  =  ( ( LSpan `  W
) `  { v } )  \/  T  =  { ( 0g `  W ) } ) )
2423ord 366 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  ( -.  T  =  ( ( LSpan `  W ) `  {
v } )  ->  T  =  { ( 0g `  W ) } ) )
2524necon1ad 2513 . . . . . . . . 9  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  ( T  =/= 
{ ( 0g `  W ) }  ->  T  =  ( ( LSpan `  W ) `  {
v } ) ) )
2615, 25mpd 14 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  =  ( ( LSpan `  W ) `  { v } ) )
2726ex 423 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( Base `  W
)  /\  v  =/=  ( 0g `  W ) ) )  ->  ( T  C_  ( ( LSpan `  W ) `  {
v } )  ->  T  =  ( ( LSpan `  W ) `  { v } ) ) )
28 eqimss 3230 . . . . . . 7  |-  ( T  =  ( ( LSpan `  W ) `  {
v } )  ->  T  C_  ( ( LSpan `  W ) `  {
v } ) )
2927, 28impbid1 194 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( Base `  W
)  /\  v  =/=  ( 0g `  W ) ) )  ->  ( T  C_  ( ( LSpan `  W ) `  {
v } )  <->  T  =  ( ( LSpan `  W
) `  { v } ) ) )
3029ex 423 . . . . 5  |-  ( ph  ->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  ( 0g `  W ) )  ->  ( T  C_  ( ( LSpan `  W
) `  { v } )  <->  T  =  ( ( LSpan `  W
) `  { v } ) ) ) )
3112, 30syl5bi 208 . . . 4  |-  ( ph  ->  ( v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } )  ->  ( T  C_  ( ( LSpan `  W
) `  { v } )  <->  T  =  ( ( LSpan `  W
) `  { v } ) ) ) )
32 sseq2 3200 . . . . . 6  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  C_  U  <->  T 
C_  ( ( LSpan `  W ) `  {
v } ) ) )
33 eqeq2 2292 . . . . . 6  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  =  U  <-> 
T  =  ( (
LSpan `  W ) `  { v } ) ) )
3432, 33bibi12d 312 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( T  C_  U 
<->  T  =  U )  <-> 
( T  C_  (
( LSpan `  W ) `  { v } )  <-> 
T  =  ( (
LSpan `  W ) `  { v } ) ) ) )
3534biimprcd 216 . . . 4  |-  ( ( T  C_  ( ( LSpan `  W ) `  { v } )  <-> 
T  =  ( (
LSpan `  W ) `  { v } ) )  ->  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  C_  U  <->  T  =  U ) ) )
3631, 35syl6 29 . . 3  |-  ( ph  ->  ( v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } )  ->  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  C_  U  <->  T  =  U ) ) ) )
3736rexlimdv 2666 . 2  |-  ( ph  ->  ( E. v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) U  =  ( ( LSpan `  W ) `  { v } )  ->  ( T  C_  U 
<->  T  =  U ) ) )
3811, 37mpd 14 1  |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855  LSAtomsclsa 29164
This theorem is referenced by:  lsatcmp2  29194  lsatel  29195  lsatnem0  29235  dvh2dimatN  31630
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-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-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-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-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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  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-lvec 15856  df-lsatoms 29166
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