Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcvexchlem5 Structured version   Visualization version   Unicode version

Theorem lcvexchlem5 32648
Description: Lemma for lcvexch 32649. (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lcvexch.s  |-  S  =  ( LSubSp `  W )
lcvexch.p  |-  .(+)  =  (
LSSum `  W )
lcvexch.c  |-  C  =  (  <oLL  `  W )
lcvexch.w  |-  ( ph  ->  W  e.  LMod )
lcvexch.t  |-  ( ph  ->  T  e.  S )
lcvexch.u  |-  ( ph  ->  U  e.  S )
lcvexch.g  |-  ( ph  ->  ( T  i^i  U
) C U )
Assertion
Ref Expression
lcvexchlem5  |-  ( ph  ->  T C ( T 
.(+)  U ) )

Proof of Theorem lcvexchlem5
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcvexch.s . . . 4  |-  S  =  ( LSubSp `  W )
2 lcvexch.c . . . 4  |-  C  =  (  <oLL  `  W )
3 lcvexch.w . . . 4  |-  ( ph  ->  W  e.  LMod )
4 lcvexch.t . . . . 5  |-  ( ph  ->  T  e.  S )
5 lcvexch.u . . . . 5  |-  ( ph  ->  U  e.  S )
61lssincl 18236 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
73, 4, 5, 6syl3anc 1276 . . . 4  |-  ( ph  ->  ( T  i^i  U
)  e.  S )
8 lcvexch.g . . . 4  |-  ( ph  ->  ( T  i^i  U
) C U )
91, 2, 3, 7, 5, 8lcvpss 32634 . . 3  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
10 lcvexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
111, 10, 2, 3, 4, 5lcvexchlem1 32644 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
129, 11mpbird 240 . 2  |-  ( ph  ->  T  C.  ( T  .(+) 
U ) )
13 simp3l 1042 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  C_  s
)
14 ssrin 3668 . . . . . . . 8  |-  ( T 
C_  s  ->  ( T  i^i  U )  C_  ( s  i^i  U
) )
1513, 14syl 17 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U )  C_  (
s  i^i  U )
)
16 inss2 3664 . . . . . . 7  |-  ( s  i^i  U )  C_  U
1715, 16jctir 545 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( ( T  i^i  U )  C_  ( s  i^i  U
)  /\  ( s  i^i  U )  C_  U
) )
1883ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U ) C U )
191, 2, 3, 7, 5lcvbr3 32633 . . . . . . . . . 10  |-  ( ph  ->  ( ( T  i^i  U ) C U  <->  ( ( T  i^i  U )  C.  U  /\  A. r  e.  S  ( ( ( T  i^i  U ) 
C_  r  /\  r  C_  U )  ->  (
r  =  ( T  i^i  U )  \/  r  =  U ) ) ) ) )
2019adantr 471 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S )  ->  (
( T  i^i  U
) C U  <->  ( ( T  i^i  U )  C.  U  /\  A. r  e.  S  ( ( ( T  i^i  U ) 
C_  r  /\  r  C_  U )  ->  (
r  =  ( T  i^i  U )  \/  r  =  U ) ) ) ) )
213adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
22 simpr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
235adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  U  e.  S )
241lssincl 18236 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  U  e.  S )  ->  (
s  i^i  U )  e.  S )
2521, 22, 23, 24syl3anc 1276 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  i^i  U )  e.  S )
26 sseq2 3465 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
( T  i^i  U
)  C_  r  <->  ( T  i^i  U )  C_  (
s  i^i  U )
) )
27 sseq1 3464 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  C_  U  <->  ( s  i^i  U )  C_  U
) )
2826, 27anbi12d 722 . . . . . . . . . . . . 13  |-  ( r  =  ( s  i^i 
U )  ->  (
( ( T  i^i  U )  C_  r  /\  r  C_  U )  <->  ( ( T  i^i  U )  C_  ( s  i^i  U
)  /\  ( s  i^i  U )  C_  U
) ) )
29 eqeq1 2465 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  ( T  i^i  U )  <->  ( s  i^i  U )  =  ( T  i^i  U ) ) )
30 eqeq1 2465 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  U  <->  ( s  i^i  U )  =  U ) )
3129, 30orbi12d 721 . . . . . . . . . . . . 13  |-  ( r  =  ( s  i^i 
U )  ->  (
( r  =  ( T  i^i  U )  \/  r  =  U )  <->  ( ( s  i^i  U )  =  ( T  i^i  U
)  \/  ( s  i^i  U )  =  U ) ) )
3228, 31imbi12d 326 . . . . . . . . . . . 12  |-  ( r  =  ( s  i^i 
U )  ->  (
( ( ( T  i^i  U )  C_  r  /\  r  C_  U
)  ->  ( r  =  ( T  i^i  U )  \/  r  =  U ) )  <->  ( (
( T  i^i  U
)  C_  ( s  i^i  U )  /\  (
s  i^i  U )  C_  U )  ->  (
( s  i^i  U
)  =  ( T  i^i  U )  \/  ( s  i^i  U
)  =  U ) ) ) )
3332rspcv 3157 . . . . . . . . . . 11  |-  ( ( s  i^i  U )  e.  S  ->  ( A. r  e.  S  ( ( ( T  i^i  U )  C_  r  /\  r  C_  U
)  ->  ( r  =  ( T  i^i  U )  \/  r  =  U ) )  -> 
( ( ( T  i^i  U )  C_  ( s  i^i  U
)  /\  ( s  i^i  U )  C_  U
)  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  \/  ( s  i^i  U )  =  U ) ) ) )
3425, 33syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  S )  ->  ( A. r  e.  S  ( ( ( T  i^i  U )  C_  r  /\  r  C_  U
)  ->  ( r  =  ( T  i^i  U )  \/  r  =  U ) )  -> 
( ( ( T  i^i  U )  C_  ( s  i^i  U
)  /\  ( s  i^i  U )  C_  U
)  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  \/  ( s  i^i  U )  =  U ) ) ) )
3534adantld 473 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S )  ->  (
( ( T  i^i  U )  C.  U  /\  A. r  e.  S  ( ( ( T  i^i  U )  C_  r  /\  r  C_  U )  -> 
( r  =  ( T  i^i  U )  \/  r  =  U ) ) )  -> 
( ( ( T  i^i  U )  C_  ( s  i^i  U
)  /\  ( s  i^i  U )  C_  U
)  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  \/  ( s  i^i  U )  =  U ) ) ) )
3620, 35sylbid 223 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S )  ->  (
( T  i^i  U
) C U  -> 
( ( ( T  i^i  U )  C_  ( s  i^i  U
)  /\  ( s  i^i  U )  C_  U
)  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  \/  ( s  i^i  U )  =  U ) ) ) )
37363adant3 1034 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( ( T  i^i  U ) C U  ->  ( (
( T  i^i  U
)  C_  ( s  i^i  U )  /\  (
s  i^i  U )  C_  U )  ->  (
( s  i^i  U
)  =  ( T  i^i  U )  \/  ( s  i^i  U
)  =  U ) ) ) )
3818, 37mpd 15 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( T  i^i  U
)  C_  ( s  i^i  U )  /\  (
s  i^i  U )  C_  U )  ->  (
( s  i^i  U
)  =  ( T  i^i  U )  \/  ( s  i^i  U
)  =  U ) ) )
3917, 38mpd 15 . . . . 5  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  \/  ( s  i^i  U )  =  U ) )
40 oveq1 6321 . . . . . . 7  |-  ( ( s  i^i  U )  =  ( T  i^i  U )  ->  ( (
s  i^i  U )  .(+)  T )  =  ( ( T  i^i  U
)  .(+)  T ) )
4133ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  W  e.  LMod )
4243ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  e.  S )
4353ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  U  e.  S )
44 simp2 1015 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  s  e.  S )
45 simp3r 1043 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  s  C_  ( T  .(+)  U ) )
461, 10, 2, 41, 42, 43, 44, 13, 45lcvexchlem3 32646 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  .(+)  T )  =  s )
471lsssssubg 18229 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
483, 47syl 17 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4948, 7sseldd 3444 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  e.  (SubGrp `  W ) )
5048, 4sseldd 3444 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
51 inss1 3663 . . . . . . . . . . 11  |-  ( T  i^i  U )  C_  T
5251a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  C_  T )
5310lsmss1 17364 . . . . . . . . . 10  |-  ( ( ( T  i^i  U
)  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )  /\  ( T  i^i  U
)  C_  T )  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
5449, 50, 52, 53syl3anc 1276 . . . . . . . . 9  |-  ( ph  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
55543ad2ant1 1035 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
5646, 55eqeq12d 2476 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( s  i^i  U
)  .(+)  T )  =  ( ( T  i^i  U )  .(+)  T )  <->  s  =  T ) )
5740, 56syl5ib 227 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  ->  s  =  T ) )
58 oveq1 6321 . . . . . . 7  |-  ( ( s  i^i  U )  =  U  ->  (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T ) )
59 lmodabl 18183 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
603, 59syl 17 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
6148, 5sseldd 3444 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
6210lsmcom 17544 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W ) )  -> 
( U  .(+)  T )  =  ( T  .(+)  U ) )
6360, 61, 50, 62syl3anc 1276 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  T )  =  ( T  .(+)  U ) )
64633ad2ant1 1035 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( U  .(+) 
T )  =  ( T  .(+)  U )
)
6546, 64eqeq12d 2476 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T )  <-> 
s  =  ( T 
.(+)  U ) ) )
6658, 65syl5ib 227 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  U  ->  s  =  ( T  .(+)  U ) ) )
6757, 66orim12d 854 . . . . 5  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( s  i^i  U
)  =  ( T  i^i  U )  \/  ( s  i^i  U
)  =  U )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) )
6839, 67mpd 15 . . . 4  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) )
69683exp 1214 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) ) )
7069ralrimiv 2811 . 2  |-  ( ph  ->  A. s  e.  S  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) )
711, 10lsmcl 18354 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
723, 4, 5, 71syl3anc 1276 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
731, 2, 3, 4, 72lcvbr3 32633 . 2  |-  ( ph  ->  ( T C ( T  .(+)  U )  <->  ( T  C.  ( T  .(+) 
U )  /\  A. s  e.  S  (
( T  C_  s  /\  s  C_  ( T 
.(+)  U ) )  -> 
( s  =  T  \/  s  =  ( T  .(+)  U )
) ) ) ) )
7412, 70, 73mpbir2and 938 1  |-  ( ph  ->  T C ( T 
.(+)  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   A.wral 2748    i^i cin 3414    C_ wss 3415    C. wpss 3416   class class class wbr 4415   ` cfv 5600  (class class class)co 6314  SubGrpcsubg 16859   LSSumclsm 17334   Abelcabl 17479   LModclmod 18139   LSubSpclss 18203    <oLL clcv 32628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-tpos 6998  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-subg 16862  df-cntz 17019  df-oppg 17045  df-lsm 17336  df-cmn 17480  df-abl 17481  df-mgp 17772  df-ur 17784  df-ring 17830  df-lmod 18141  df-lss 18204  df-lcv 32629
This theorem is referenced by:  lcvexch  32649
  Copyright terms: Public domain W3C validator