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Theorem lsmfgcl 35978
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 2462 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
8 eqid 2462 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
95, 6, 7, 8islssfg2 35975 . . . . 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 671 . . . 4  |-  ( ph  ->  ( D  e. LFinGen  <->  E. a  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  a
)  =  A ) )
112, 10mpbid 215 . . 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 35975 . . . . . . . . 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 671 . . . . . . . 8  |-  ( ph  ->  ( E  e. LFinGen  <->  E. b  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  b
)  =  B ) )
1712, 16mpbid 215 . . . . . . 7  |-  ( ph  ->  E. b  e.  ( ~P ( Base `  W
)  i^i  Fin )
( ( LSpan `  W
) `  b )  =  B )
1817adantr 471 . . . . . 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 3664 . . . . . . . . . . . . . . 15  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
2019sseli 3440 . . . . . . . . . . . . . 14  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  e.  ~P ( Base `  W
) )
2120elpwid 3973 . . . . . . . . . . . . 13  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  C_  ( Base `  W
) )
2219sseli 3440 . . . . . . . . . . . . . 14  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  e.  ~P ( Base `  W
) )
2322elpwid 3973 . . . . . . . . . . . . 13  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  C_  ( Base `  W
) )
24 lsmfgcl.p . . . . . . . . . . . . . 14  |-  .(+)  =  (
LSSum `  W )
258, 7, 24lsmsp2 18365 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  a  C_  ( Base `  W
)  /\  b  C_  ( Base `  W )
)  ->  ( (
( LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) )  =  ( ( LSpan `  W
) `  ( a  u.  b ) ) )
263, 21, 23, 25syl3an 1318 . . . . . . . . . . . 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 ) ) )
27263expb 1216 . . . . . . . . . . 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
) ) )
2827oveq2d 6336 . . . . . . . . . 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 ) ) ) )
293adantr 471 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  W  e.  LMod )
30 unss 3620 . . . . . . . . . . . . . 14  |-  ( ( a  C_  ( Base `  W )  /\  b  C_  ( Base `  W
) )  <->  ( a  u.  b )  C_  ( Base `  W ) )
3130biimpi 199 . . . . . . . . . . . . 13  |-  ( ( a  C_  ( Base `  W )  /\  b  C_  ( Base `  W
) )  ->  (
a  u.  b ) 
C_  ( Base `  W
) )
3221, 23, 31syl2an 484 . . . . . . . . . . . 12  |-  ( ( a  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( a  u.  b )  C_  ( Base `  W ) )
3332adantl 472 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  (
a  u.  b ) 
C_  ( Base `  W
) )
34 inss2 3665 . . . . . . . . . . . . . 14  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
Fin
3534sseli 3440 . . . . . . . . . . . . 13  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  e.  Fin )
3634sseli 3440 . . . . . . . . . . . . 13  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  e.  Fin )
37 unfi 7869 . . . . . . . . . . . . 13  |-  ( ( a  e.  Fin  /\  b  e.  Fin )  ->  ( a  u.  b
)  e.  Fin )
3835, 36, 37syl2an 484 . . . . . . . . . . . 12  |-  ( ( a  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( a  u.  b )  e.  Fin )
3938adantl 472 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  (
a  u.  b )  e.  Fin )
40 eqid 2462 . . . . . . . . . . . 12  |-  ( Ws  ( ( LSpan `  W ) `  ( a  u.  b
) ) )  =  ( Ws  ( ( LSpan `  W ) `  (
a  u.  b ) ) )
417, 8, 40islssfgi 35976 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  (
a  u.  b ) 
C_  ( Base `  W
)  /\  ( a  u.  b )  e.  Fin )  ->  ( Ws  ( (
LSpan `  W ) `  ( a  u.  b
) ) )  e. LFinGen )
4229, 33, 39, 41syl3anc 1276 . . . . . . . . . 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 )
4328, 42eqeltrd 2540 . . . . . . . . 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 )
4443anassrs 658 . . . . . . . 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 )
45 oveq2 6328 . . . . . . . . . 10  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( ( (
LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) )  =  ( ( ( LSpan `  W ) `  a
)  .(+)  B ) )
4645oveq2d 6336 . . . . . . . . 9  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) ) )  =  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  B ) ) )
4746eleq1d 2524 . . . . . . . 8  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( ( Ws  ( ( ( LSpan `  W
) `  a )  .(+)  ( ( LSpan `  W
) `  b )
) )  e. LFinGen  <->  ( Ws  (
( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen ) )
4844, 47syl5ibcom 228 . . . . . . 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 ) )
4948rexlimdva 2891 . . . . . 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 ) )
5018, 49mpd 15 . . . . 5  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( Ws  (
( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen )
51 oveq1 6327 . . . . . . 7  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( ( (
LSpan `  W ) `  a )  .(+)  B )  =  ( A  .(+)  B ) )
5251oveq2d 6336 . . . . . 6  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  B ) )  =  ( Ws  ( A  .(+)  B )
) )
5352eleq1d 2524 . . . . 5  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( ( Ws  ( ( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen  <->  ( Ws  ( A  .(+)  B ) )  e. LFinGen ) )
5450, 53syl5ibcom 228 . . . 4  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( (
( LSpan `  W ) `  a )  =  A  ->  ( Ws  ( A 
.(+)  B ) )  e. LFinGen ) )
5554rexlimdva 2891 . . 3  |-  ( ph  ->  ( E. a  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  a
)  =  A  -> 
( Ws  ( A  .(+)  B ) )  e. LFinGen )
)
5611, 55mpd 15 . 2  |-  ( ph  ->  ( Ws  ( A  .(+)  B ) )  e. LFinGen )
571, 56syl5eqel 2544 1  |-  ( ph  ->  F  e. LFinGen )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   E.wrex 2750    u. cun 3414    i^i cin 3415    C_ wss 3416   ~Pcpw 3963   ` cfv 5605  (class class class)co 6320   Fincfn 7600   Basecbs 15176   ↾s cress 15177   LSSumclsm 17341   LModclmod 18146   LSubSpclss 18210   LSpanclspn 18249  LFinGenclfig 35971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-oadd 7217  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-sca 15261  df-vsca 15262  df-0g 15395  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-submnd 16638  df-grp 16728  df-minusg 16729  df-sbg 16730  df-subg 16869  df-cntz 17026  df-lsm 17343  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-lmod 18148  df-lss 18211  df-lsp 18250  df-lfig 35972
This theorem is referenced by:  lmhmfgsplit  35990
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