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Theorem lsatcmp 35180
Description: If two atoms are comparable, they are equal. (atsseq 27407 analog.) TODO: can lspsncmp 17898 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 17888 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
42, 3syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
5 eqid 2396 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
6 eqid 2396 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
7 eqid 2396 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
8 lsatcmp.a . . . . 5  |-  A  =  (LSAtoms `  W )
95, 6, 7, 8islsat 35168 . . . 4  |-  ( W  e.  LMod  ->  ( U  e.  A  <->  E. v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) U  =  ( ( LSpan `  W ) `  { v } ) ) )
104, 9syl 16 . . 3  |-  ( ph  ->  ( U  e.  A  <->  E. v  e.  ( (
Base `  W )  \  { ( 0g `  W ) } ) U  =  ( (
LSpan `  W ) `  { v } ) ) )
111, 10mpbid 210 . 2  |-  ( ph  ->  E. v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) U  =  ( ( LSpan `  W ) `  { v } ) )
12 eldifsn 4086 . . . . 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 35176 . . . . . . . . . 10  |-  ( ph  ->  T  =/=  { ( 0g `  W ) } )
1514ad2antrr 723 . . . . . . . . 9  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  =/=  {
( 0g `  W
) } )
162ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  W  e.  LVec )
17 eqid 2396 . . . . . . . . . . . . . 14  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1817, 8, 4, 13lsatlssel 35174 . . . . . . . . . . . . 13  |-  ( ph  ->  T  e.  ( LSubSp `  W ) )
1918ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  e.  (
LSubSp `  W ) )
20 simplrl 759 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  v  e.  (
Base `  W )
)
21 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  C_  (
( LSpan `  W ) `  { v } ) )
225, 7, 17, 6lspsnat 17927 . . . . . . . . . . . 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 1229 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  ( T  =  ( ( LSpan `  W
) `  { v } )  \/  T  =  { ( 0g `  W ) } ) )
2423ord 375 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  ( -.  T  =  ( ( LSpan `  W ) `  {
v } )  ->  T  =  { ( 0g `  W ) } ) )
2524necon1ad 2612 . . . . . . . . 9  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  ( T  =/= 
{ ( 0g `  W ) }  ->  T  =  ( ( LSpan `  W ) `  {
v } ) ) )
2615, 25mpd 15 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  v  =/=  ( 0g `  W
) ) )  /\  T  C_  ( ( LSpan `  W ) `  {
v } ) )  ->  T  =  ( ( LSpan `  W ) `  { v } ) )
2726ex 432 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( Base `  W
)  /\  v  =/=  ( 0g `  W ) ) )  ->  ( T  C_  ( ( LSpan `  W ) `  {
v } )  ->  T  =  ( ( LSpan `  W ) `  { v } ) ) )
28 eqimss 3486 . . . . . . 7  |-  ( T  =  ( ( LSpan `  W ) `  {
v } )  ->  T  C_  ( ( LSpan `  W ) `  {
v } ) )
2927, 28impbid1 203 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( Base `  W
)  /\  v  =/=  ( 0g `  W ) ) )  ->  ( T  C_  ( ( LSpan `  W ) `  {
v } )  <->  T  =  ( ( LSpan `  W
) `  { v } ) ) )
3029ex 432 . . . . 5  |-  ( ph  ->  ( ( v  e.  ( Base `  W
)  /\  v  =/=  ( 0g `  W ) )  ->  ( T  C_  ( ( LSpan `  W
) `  { v } )  <->  T  =  ( ( LSpan `  W
) `  { v } ) ) ) )
3112, 30syl5bi 217 . . . 4  |-  ( ph  ->  ( v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } )  ->  ( T  C_  ( ( LSpan `  W
) `  { v } )  <->  T  =  ( ( LSpan `  W
) `  { v } ) ) ) )
32 sseq2 3456 . . . . . 6  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  C_  U  <->  T 
C_  ( ( LSpan `  W ) `  {
v } ) ) )
33 eqeq2 2411 . . . . . 6  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  =  U  <-> 
T  =  ( (
LSpan `  W ) `  { v } ) ) )
3432, 33bibi12d 319 . . . . 5  |-  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( T  C_  U 
<->  T  =  U )  <-> 
( T  C_  (
( LSpan `  W ) `  { v } )  <-> 
T  =  ( (
LSpan `  W ) `  { v } ) ) ) )
3534biimprcd 225 . . . 4  |-  ( ( T  C_  ( ( LSpan `  W ) `  { v } )  <-> 
T  =  ( (
LSpan `  W ) `  { v } ) )  ->  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  C_  U  <->  T  =  U ) ) )
3631, 35syl6 33 . . 3  |-  ( ph  ->  ( v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } )  ->  ( U  =  ( ( LSpan `  W ) `  {
v } )  -> 
( T  C_  U  <->  T  =  U ) ) ) )
3736rexlimdv 2886 . 2  |-  ( ph  ->  ( E. v  e.  ( ( Base `  W
)  \  { ( 0g `  W ) } ) U  =  ( ( LSpan `  W ) `  { v } )  ->  ( T  C_  U 
<->  T  =  U ) ) )
3811, 37mpd 15 1  |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1836    =/= wne 2591   E.wrex 2747    \ cdif 3403    C_ wss 3406   {csn 3961   ` cfv 5513   Basecbs 14657   0gc0g 14870   LModclmod 17648   LSubSpclss 17714   LSpanclspn 17753   LVecclvec 17884  LSAtomsclsa 35151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-tpos 6895  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-2 10533  df-3 10534  df-ndx 14660  df-slot 14661  df-base 14662  df-sets 14663  df-ress 14664  df-plusg 14738  df-mulr 14739  df-0g 14872  df-mgm 16012  df-sgrp 16051  df-mnd 16061  df-grp 16197  df-minusg 16198  df-sbg 16199  df-cmn 16940  df-abl 16941  df-mgp 17278  df-ur 17290  df-ring 17336  df-oppr 17408  df-dvdsr 17426  df-unit 17427  df-invr 17457  df-drng 17534  df-lmod 17650  df-lss 17715  df-lsp 17754  df-lvec 17885  df-lsatoms 35153
This theorem is referenced by:  lsatcmp2  35181  lsatel  35182  lsatnem0  35222  dvh2dimatN  37619
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