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

Theorem lcvexchlem5 34864
Description: Lemma for lcvexch 34865. (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 17737 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
73, 4, 5, 6syl3anc 1228 . . . 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 34850 . . 3  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
10 lcvexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
111, 10, 2, 3, 4, 5lcvexchlem1 34860 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
129, 11mpbird 232 . 2  |-  ( ph  ->  T  C.  ( T  .(+) 
U ) )
13 simp3l 1024 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  C_  s
)
14 ssrin 3719 . . . . . . . 8  |-  ( T 
C_  s  ->  ( T  i^i  U )  C_  ( s  i^i  U
) )
1513, 14syl 16 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U )  C_  (
s  i^i  U )
)
16 inss2 3715 . . . . . . 7  |-  ( s  i^i  U )  C_  U
1715, 16jctir 538 . . . . . 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 1017 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U ) C U )
191, 2, 3, 7, 5lcvbr3 34849 . . . . . . . . . 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 465 . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
22 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
235adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  U  e.  S )
241lssincl 17737 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  U  e.  S )  ->  (
s  i^i  U )  e.  S )
2521, 22, 23, 24syl3anc 1228 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  i^i  U )  e.  S )
26 sseq2 3521 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
( T  i^i  U
)  C_  r  <->  ( T  i^i  U )  C_  (
s  i^i  U )
) )
27 sseq1 3520 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  C_  U  <->  ( s  i^i  U )  C_  U
) )
2826, 27anbi12d 710 . . . . . . . . . . . . 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 2461 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  ( T  i^i  U )  <->  ( s  i^i  U )  =  ( T  i^i  U ) ) )
30 eqeq1 2461 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  U  <->  ( s  i^i  U )  =  U ) )
3129, 30orbi12d 709 . . . . . . . . . . . . 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 320 . . . . . . . . . . . 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 3206 . . . . . . . . . . 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 16 . . . . . . . . . 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 467 . . . . . . . . 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 215 . . . . . . . 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 1016 . . . . . . 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 6303 . . . . . . 7  |-  ( ( s  i^i  U )  =  ( T  i^i  U )  ->  ( (
s  i^i  U )  .(+)  T )  =  ( ( T  i^i  U
)  .(+)  T ) )
4133ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  W  e.  LMod )
4243ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  e.  S )
4353ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  U  e.  S )
44 simp2 997 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  s  e.  S )
45 simp3r 1025 . . . . . . . . 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 34862 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  .(+)  T )  =  s )
471lsssssubg 17730 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
483, 47syl 16 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4948, 7sseldd 3500 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  e.  (SubGrp `  W ) )
5048, 4sseldd 3500 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
51 inss1 3714 . . . . . . . . . . 11  |-  ( T  i^i  U )  C_  T
5251a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  C_  T )
5310lsmss1 16810 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
55543ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
5646, 55eqeq12d 2479 . . . . . . 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 219 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  ->  s  =  T ) )
58 oveq1 6303 . . . . . . 7  |-  ( ( s  i^i  U )  =  U  ->  (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T ) )
59 lmodabl 17683 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
603, 59syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
6148, 5sseldd 3500 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
6210lsmcom 16990 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W ) )  -> 
( U  .(+)  T )  =  ( T  .(+)  U ) )
6360, 61, 50, 62syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  T )  =  ( T  .(+)  U ) )
64633ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( U  .(+) 
T )  =  ( T  .(+)  U )
)
6546, 64eqeq12d 2479 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T )  <-> 
s  =  ( T 
.(+)  U ) ) )
6658, 65syl5ib 219 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  U  ->  s  =  ( T  .(+)  U ) ) )
6757, 66orim12d 838 . . . . 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 1195 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) ) )
7069ralrimiv 2869 . 2  |-  ( ph  ->  A. s  e.  S  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) )
711, 10lsmcl 17855 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
723, 4, 5, 71syl3anc 1228 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
731, 2, 3, 4, 72lcvbr3 34849 . 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 922 1  |-  ( ph  ->  T C ( T 
.(+)  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    i^i cin 3470    C_ wss 3471    C. wpss 3472   class class class wbr 4456   ` cfv 5594  (class class class)co 6296  SubGrpcsubg 16321   LSSumclsm 16780   Abelcabl 16925   LModclmod 17638   LSubSpclss 17704    <oLL clcv 34844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-oppg 16507  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-lmod 17640  df-lss 17705  df-lcv 34845
This theorem is referenced by:  lcvexch  34865
  Copyright terms: Public domain W3C validator