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Theorem lsmfgcl 27275
Description: The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
lsmfgcl.u  |-  U  =  ( LSubSp `  W )
lsmfgcl.p  |-  .(+)  =  (
LSSum `  W )
lsmfgcl.d  |-  D  =  ( Ws  A )
lsmfgcl.e  |-  E  =  ( Ws  B )
lsmfgcl.f  |-  F  =  ( Ws  ( A  .(+)  B ) )
lsmfgcl.w  |-  ( ph  ->  W  e.  LMod )
lsmfgcl.a  |-  ( ph  ->  A  e.  U )
lsmfgcl.b  |-  ( ph  ->  B  e.  U )
lsmfgcl.df  |-  ( ph  ->  D  e. LFinGen )
lsmfgcl.ef  |-  ( ph  ->  E  e. LFinGen )
Assertion
Ref Expression
lsmfgcl  |-  ( ph  ->  F  e. LFinGen )

Proof of Theorem lsmfgcl
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmfgcl.f . 2  |-  F  =  ( Ws  ( A  .(+)  B ) )
2 lsmfgcl.df . . . 4  |-  ( ph  ->  D  e. LFinGen )
3 lsmfgcl.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
4 lsmfgcl.a . . . . 5  |-  ( ph  ->  A  e.  U )
5 lsmfgcl.d . . . . . 6  |-  D  =  ( Ws  A )
6 lsmfgcl.u . . . . . 6  |-  U  =  ( LSubSp `  W )
7 eqid 2296 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
8 eqid 2296 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
95, 6, 7, 8islssfg2 27272 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  U )  ->  ( D  e. LFinGen  <->  E. a  e.  ( ~P ( Base `  W
)  i^i  Fin )
( ( LSpan `  W
) `  a )  =  A ) )
103, 4, 9syl2anc 642 . . . 4  |-  ( ph  ->  ( D  e. LFinGen  <->  E. a  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  a
)  =  A ) )
112, 10mpbid 201 . . 3  |-  ( ph  ->  E. a  e.  ( ~P ( Base `  W
)  i^i  Fin )
( ( LSpan `  W
) `  a )  =  A )
12 lsmfgcl.ef . . . . . . . 8  |-  ( ph  ->  E  e. LFinGen )
13 lsmfgcl.b . . . . . . . . 9  |-  ( ph  ->  B  e.  U )
14 lsmfgcl.e . . . . . . . . . 10  |-  E  =  ( Ws  B )
1514, 6, 7, 8islssfg2 27272 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  B  e.  U )  ->  ( E  e. LFinGen  <->  E. b  e.  ( ~P ( Base `  W
)  i^i  Fin )
( ( LSpan `  W
) `  b )  =  B ) )
163, 13, 15syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( E  e. LFinGen  <->  E. b  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  b
)  =  B ) )
1712, 16mpbid 201 . . . . . . 7  |-  ( ph  ->  E. b  e.  ( ~P ( Base `  W
)  i^i  Fin )
( ( LSpan `  W
) `  b )  =  B )
1817adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  E. b  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  b
)  =  B )
19 inss1 3402 . . . . . . . . . . . . . . 15  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
2019sseli 3189 . . . . . . . . . . . . . 14  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  e.  ~P ( Base `  W
) )
21 elpwi 3646 . . . . . . . . . . . . . 14  |-  ( a  e.  ~P ( Base `  W )  ->  a  C_  ( Base `  W
) )
2220, 21syl 15 . . . . . . . . . . . . 13  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  C_  ( Base `  W
) )
2319sseli 3189 . . . . . . . . . . . . . 14  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  e.  ~P ( Base `  W
) )
24 elpwi 3646 . . . . . . . . . . . . . 14  |-  ( b  e.  ~P ( Base `  W )  ->  b  C_  ( Base `  W
) )
2523, 24syl 15 . . . . . . . . . . . . 13  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  C_  ( Base `  W
) )
26 lsmfgcl.p . . . . . . . . . . . . . 14  |-  .(+)  =  (
LSSum `  W )
278, 7, 26lsmsp2 15856 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  a  C_  ( Base `  W
)  /\  b  C_  ( Base `  W )
)  ->  ( (
( LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) )  =  ( ( LSpan `  W
) `  ( a  u.  b ) ) )
283, 22, 25, 27syl3an 1224 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( (
( LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) )  =  ( ( LSpan `  W
) `  ( a  u.  b ) ) )
29283expb 1152 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  (
( ( LSpan `  W
) `  a )  .(+)  ( ( LSpan `  W
) `  b )
)  =  ( (
LSpan `  W ) `  ( a  u.  b
) ) )
3029oveq2d 5890 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  ( Ws  ( ( ( LSpan `  W ) `  a
)  .(+)  ( ( LSpan `  W ) `  b
) ) )  =  ( Ws  ( ( LSpan `  W ) `  (
a  u.  b ) ) ) )
313adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  W  e.  LMod )
32 unss 3362 . . . . . . . . . . . . . 14  |-  ( ( a  C_  ( Base `  W )  /\  b  C_  ( Base `  W
) )  <->  ( a  u.  b )  C_  ( Base `  W ) )
3332biimpi 186 . . . . . . . . . . . . 13  |-  ( ( a  C_  ( Base `  W )  /\  b  C_  ( Base `  W
) )  ->  (
a  u.  b ) 
C_  ( Base `  W
) )
3422, 25, 33syl2an 463 . . . . . . . . . . . 12  |-  ( ( a  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( a  u.  b )  C_  ( Base `  W ) )
3534adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  (
a  u.  b ) 
C_  ( Base `  W
) )
36 inss2 3403 . . . . . . . . . . . . . 14  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
Fin
3736sseli 3189 . . . . . . . . . . . . 13  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  e.  Fin )
3836sseli 3189 . . . . . . . . . . . . 13  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  e.  Fin )
39 unfi 7140 . . . . . . . . . . . . 13  |-  ( ( a  e.  Fin  /\  b  e.  Fin )  ->  ( a  u.  b
)  e.  Fin )
4037, 38, 39syl2an 463 . . . . . . . . . . . 12  |-  ( ( a  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( a  u.  b )  e.  Fin )
4140adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  (
a  u.  b )  e.  Fin )
42 eqid 2296 . . . . . . . . . . . 12  |-  ( Ws  ( ( LSpan `  W ) `  ( a  u.  b
) ) )  =  ( Ws  ( ( LSpan `  W ) `  (
a  u.  b ) ) )
437, 8, 42islssfgi 27273 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  (
a  u.  b ) 
C_  ( Base `  W
)  /\  ( a  u.  b )  e.  Fin )  ->  ( Ws  ( (
LSpan `  W ) `  ( a  u.  b
) ) )  e. LFinGen )
4431, 35, 41, 43syl3anc 1182 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  ( Ws  ( ( LSpan `  W
) `  ( a  u.  b ) ) )  e. LFinGen )
4530, 44eqeltrd 2370 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  ( Ws  ( ( ( LSpan `  W ) `  a
)  .(+)  ( ( LSpan `  W ) `  b
) ) )  e. LFinGen )
4645anassrs 629 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( ~P ( Base `  W )  i^i  Fin ) )  /\  b  e.  ( ~P ( Base `  W )  i^i  Fin ) )  ->  ( Ws  ( ( ( LSpan `  W ) `  a
)  .(+)  ( ( LSpan `  W ) `  b
) ) )  e. LFinGen )
47 oveq2 5882 . . . . . . . . . 10  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( ( (
LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) )  =  ( ( ( LSpan `  W ) `  a
)  .(+)  B ) )
4847oveq2d 5890 . . . . . . . . 9  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) ) )  =  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  B ) ) )
4948eleq1d 2362 . . . . . . . 8  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( ( Ws  ( ( ( LSpan `  W
) `  a )  .(+)  ( ( LSpan `  W
) `  b )
) )  e. LFinGen  <->  ( Ws  (
( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen ) )
5046, 49syl5ibcom 211 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( ~P ( Base `  W )  i^i  Fin ) )  /\  b  e.  ( ~P ( Base `  W )  i^i  Fin ) )  ->  (
( ( LSpan `  W
) `  b )  =  B  ->  ( Ws  ( ( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen ) )
5150rexlimdva 2680 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( E. b  e.  ( ~P ( Base `  W )  i^i  Fin ) ( (
LSpan `  W ) `  b )  =  B  ->  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  B ) )  e. LFinGen ) )
5218, 51mpd 14 . . . . 5  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( Ws  (
( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen )
53 oveq1 5881 . . . . . . 7  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( ( (
LSpan `  W ) `  a )  .(+)  B )  =  ( A  .(+)  B ) )
5453oveq2d 5890 . . . . . 6  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  B ) )  =  ( Ws  ( A  .(+)  B )
) )
5554eleq1d 2362 . . . . 5  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( ( Ws  ( ( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen  <->  ( Ws  ( A  .(+)  B ) )  e. LFinGen ) )
5652, 55syl5ibcom 211 . . . 4  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( (
( LSpan `  W ) `  a )  =  A  ->  ( Ws  ( A 
.(+)  B ) )  e. LFinGen ) )
5756rexlimdva 2680 . . 3  |-  ( ph  ->  ( E. a  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  a
)  =  A  -> 
( Ws  ( A  .(+)  B ) )  e. LFinGen )
)
5811, 57mpd 14 . 2  |-  ( ph  ->  ( Ws  ( A  .(+)  B ) )  e. LFinGen )
591, 58syl5eqel 2380 1  |-  ( ph  ->  F  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    u. cun 3163    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   ↾s cress 13165   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744  LFinGenclfig 27268
This theorem is referenced by:  lmhmfgsplit  27287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lfig 27269
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