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Theorem lsmfgcl 27149
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 2436 . . . . . 6  |-  ( LSpan `  W )  =  (
LSpan `  W )
8 eqid 2436 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
95, 6, 7, 8islssfg2 27146 . . . . 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 643 . . . 4  |-  ( ph  ->  ( D  e. LFinGen  <->  E. a  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  a
)  =  A ) )
112, 10mpbid 202 . . 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 27146 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ph  ->  ( E  e. LFinGen  <->  E. b  e.  ( ~P ( Base `  W )  i^i  Fin ) ( ( LSpan `  W ) `  b
)  =  B ) )
1712, 16mpbid 202 . . . . . . 7  |-  ( ph  ->  E. b  e.  ( ~P ( Base `  W
)  i^i  Fin )
( ( LSpan `  W
) `  b )  =  B )
1817adantr 452 . . . . . 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 3561 . . . . . . . . . . . . . . 15  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
2019sseli 3344 . . . . . . . . . . . . . 14  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  e.  ~P ( Base `  W
) )
2120elpwid 3808 . . . . . . . . . . . . 13  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  C_  ( Base `  W
) )
2219sseli 3344 . . . . . . . . . . . . . 14  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  e.  ~P ( Base `  W
) )
2322elpwid 3808 . . . . . . . . . . . . 13  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  C_  ( Base `  W
) )
24 lsmfgcl.p . . . . . . . . . . . . . 14  |-  .(+)  =  (
LSSum `  W )
258, 7, 24lsmsp2 16159 . . . . . . . . . . . . 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 1226 . . . . . . . . . . . 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 1154 . . . . . . . . . . 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 6097 . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  ( ~P ( Base `  W )  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
) )  ->  W  e.  LMod )
30 unss 3521 . . . . . . . . . . . . . 14  |-  ( ( a  C_  ( Base `  W )  /\  b  C_  ( Base `  W
) )  <->  ( a  u.  b )  C_  ( Base `  W ) )
3130biimpi 187 . . . . . . . . . . . . 13  |-  ( ( a  C_  ( Base `  W )  /\  b  C_  ( Base `  W
) )  ->  (
a  u.  b ) 
C_  ( Base `  W
) )
3221, 23, 31syl2an 464 . . . . . . . . . . . 12  |-  ( ( a  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( a  u.  b )  C_  ( Base `  W ) )
3332adantl 453 . . . . . . . . . . 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 3562 . . . . . . . . . . . . . 14  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
Fin
3534sseli 3344 . . . . . . . . . . . . 13  |-  ( a  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  a  e.  Fin )
3634sseli 3344 . . . . . . . . . . . . 13  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  ->  b  e.  Fin )
37 unfi 7374 . . . . . . . . . . . . 13  |-  ( ( a  e.  Fin  /\  b  e.  Fin )  ->  ( a  u.  b
)  e.  Fin )
3835, 36, 37syl2an 464 . . . . . . . . . . . 12  |-  ( ( a  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  b  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( a  u.  b )  e.  Fin )
3938adantl 453 . . . . . . . . . . 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 2436 . . . . . . . . . . . 12  |-  ( Ws  ( ( LSpan `  W ) `  ( a  u.  b
) ) )  =  ( Ws  ( ( LSpan `  W ) `  (
a  u.  b ) ) )
417, 8, 40islssfgi 27147 . . . . . . . . . . 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 1184 . . . . . . . . . 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 2510 . . . . . . . . 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 630 . . . . . . . 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 6089 . . . . . . . . . 10  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( ( (
LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) )  =  ( ( ( LSpan `  W ) `  a
)  .(+)  B ) )
4645oveq2d 6097 . . . . . . . . 9  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  ( (
LSpan `  W ) `  b ) ) )  =  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  B ) ) )
4746eleq1d 2502 . . . . . . . 8  |-  ( ( ( LSpan `  W ) `  b )  =  B  ->  ( ( Ws  ( ( ( LSpan `  W
) `  a )  .(+)  ( ( LSpan `  W
) `  b )
) )  e. LFinGen  <->  ( Ws  (
( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen ) )
4844, 47syl5ibcom 212 . . . . . . 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 2830 . . . . . 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 6088 . . . . . . 7  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( ( (
LSpan `  W ) `  a )  .(+)  B )  =  ( A  .(+)  B ) )
5251oveq2d 6097 . . . . . 6  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( Ws  ( ( ( LSpan `  W ) `  a )  .(+)  B ) )  =  ( Ws  ( A  .(+)  B )
) )
5352eleq1d 2502 . . . . 5  |-  ( ( ( LSpan `  W ) `  a )  =  A  ->  ( ( Ws  ( ( ( LSpan `  W
) `  a )  .(+)  B ) )  e. LFinGen  <->  ( Ws  ( A  .(+)  B ) )  e. LFinGen ) )
5450, 53syl5ibcom 212 . . . 4  |-  ( (
ph  /\  a  e.  ( ~P ( Base `  W
)  i^i  Fin )
)  ->  ( (
( LSpan `  W ) `  a )  =  A  ->  ( Ws  ( A 
.(+)  B ) )  e. LFinGen ) )
5554rexlimdva 2830 . . 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 2520 1  |-  ( ph  ->  F  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    u. cun 3318    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   ` cfv 5454  (class class class)co 6081   Fincfn 7109   Basecbs 13469   ↾s cress 13470   LSSumclsm 15268   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047  LFinGenclfig 27142
This theorem is referenced by:  lmhmfgsplit  27161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lfig 27143
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