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Theorem islss3 17040
Description: A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
islss3.x  |-  X  =  ( Ws  U )
islss3.v  |-  V  =  ( Base `  W
)
islss3.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
islss3  |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  C_  V  /\  X  e. 
LMod ) ) )

Proof of Theorem islss3
Dummy variables  a 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islss3.v . . . . 5  |-  V  =  ( Base `  W
)
2 islss3.s . . . . 5  |-  S  =  ( LSubSp `  W )
31, 2lssss 17018 . . . 4  |-  ( U  e.  S  ->  U  C_  V )
43adantl 466 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  C_  V )
5 islss3.x . . . . . . 7  |-  X  =  ( Ws  U )
65, 1ressbas2 14229 . . . . . 6  |-  ( U 
C_  V  ->  U  =  ( Base `  X
) )
76adantl 466 . . . . 5  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  =  ( Base `  X
) )
83, 7sylan2 474 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  =  ( Base `  X
) )
9 eqid 2443 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
105, 9ressplusg 14280 . . . . 5  |-  ( U  e.  S  ->  ( +g  `  W )  =  ( +g  `  X
) )
1110adantl 466 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  W )  =  ( +g  `  X
) )
12 eqid 2443 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
135, 12resssca 14316 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 466 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
15 eqid 2443 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
165, 15ressvsca 14317 . . . . 5  |-  ( U  e.  S  ->  ( .s `  W )  =  ( .s `  X
) )
1716adantl 466 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .s `  W )  =  ( .s `  X
) )
18 eqidd 2444 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
19 eqidd 2444 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  (Scalar `  W
) )  =  ( +g  `  (Scalar `  W ) ) )
20 eqidd 2444 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .r `  (Scalar `  W
) )  =  ( .r `  (Scalar `  W ) ) )
21 eqidd 2444 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 1r `  (Scalar `  W
) )  =  ( 1r `  (Scalar `  W ) ) )
2212lmodrng 16956 . . . . 5  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
2322adantr 465 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  e.  Ring )
242lsssubg 17038 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
255subggrp 15684 . . . . 5  |-  ( U  e.  (SubGrp `  W
)  ->  X  e.  Grp )
2624, 25syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  Grp )
27 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2812, 15, 27, 2lssvscl 17036 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U )
)  ->  ( x
( .s `  W
) a )  e.  U )
29283impb 1183 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  (
Base `  (Scalar `  W
) )  /\  a  e.  U )  ->  (
x ( .s `  W ) a )  e.  U )
30 simpll 753 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  W  e.  LMod )
31 simpr1 994 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  x  e.  ( Base `  (Scalar `  W
) ) )
323ad2antlr 726 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  U  C_  V
)
33 simpr2 995 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  U )
3432, 33sseldd 3357 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  V )
35 simpr3 996 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  b  e.  U )
3632, 35sseldd 3357 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  b  e.  V )
371, 9, 12, 15, 27lmodvsdi 16971 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  (
x ( .s `  W ) ( a ( +g  `  W
) b ) )  =  ( ( x ( .s `  W
) a ) ( +g  `  W ) ( x ( .s
`  W ) b ) ) )
3830, 31, 34, 36, 37syl13anc 1220 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  ( x
( .s `  W
) ( a ( +g  `  W ) b ) )  =  ( ( x ( .s `  W ) a ) ( +g  `  W ) ( x ( .s `  W
) b ) ) )
39 simpll 753 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  ->  W  e.  LMod )
40 simpr1 994 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  ->  x  e.  ( Base `  (Scalar `  W )
) )
41 simpr2 995 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  -> 
a  e.  ( Base `  (Scalar `  W )
) )
423ad2antlr 726 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  ->  U  C_  V )
43 simpr3 996 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  -> 
b  e.  U )
4442, 43sseldd 3357 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  -> 
b  e.  V )
45 eqid 2443 . . . . . 6  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
461, 9, 12, 15, 27, 45lmodvsdir 16972 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  ( Base `  (Scalar `  W
) )  /\  b  e.  V ) )  -> 
( ( x ( +g  `  (Scalar `  W ) ) a ) ( .s `  W ) b )  =  ( ( x ( .s `  W
) b ) ( +g  `  W ) ( a ( .s
`  W ) b ) ) )
4739, 40, 41, 44, 46syl13anc 1220 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  -> 
( ( x ( +g  `  (Scalar `  W ) ) a ) ( .s `  W ) b )  =  ( ( x ( .s `  W
) b ) ( +g  `  W ) ( a ( .s
`  W ) b ) ) )
48 eqid 2443 . . . . . 6  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
491, 12, 15, 27, 48lmodvsass 16973 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  ( Base `  (Scalar `  W
) )  /\  b  e.  V ) )  -> 
( ( x ( .r `  (Scalar `  W ) ) a ) ( .s `  W ) b )  =  ( x ( .s `  W ) ( a ( .s
`  W ) b ) ) )
5039, 40, 41, 44, 49syl13anc 1220 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  ( Base `  (Scalar `  W )
)  /\  b  e.  U ) )  -> 
( ( x ( .r `  (Scalar `  W ) ) a ) ( .s `  W ) b )  =  ( x ( .s `  W ) ( a ( .s
`  W ) b ) ) )
514sselda 3356 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  U
)  ->  x  e.  V )
52 eqid 2443 . . . . . . 7  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
531, 12, 15, 52lmodvs1 16976 . . . . . 6  |-  ( ( W  e.  LMod  /\  x  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) x )  =  x )
5453adantlr 714 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  V
)  ->  ( ( 1r `  (Scalar `  W
) ) ( .s
`  W ) x )  =  x )
5551, 54syldan 470 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  U
)  ->  ( ( 1r `  (Scalar `  W
) ) ( .s
`  W ) x )  =  x )
568, 11, 14, 17, 18, 19, 20, 21, 23, 26, 29, 38, 47, 50, 55islmodd 16954 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
574, 56jca 532 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  V  /\  X  e.  LMod ) )
58 simprl 755 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  C_  V )
5958, 6syl 16 . . 3  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  =  ( Base `  X ) )
60 fvex 5701 . . . . . . 7  |-  ( Base `  X )  e.  _V
6159, 60syl6eqel 2531 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  _V )
625, 12resssca 14316 . . . . . 6  |-  ( U  e.  _V  ->  (Scalar `  W )  =  (Scalar `  X ) )
6361, 62syl 16 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
(Scalar `  W )  =  (Scalar `  X )
)
6463eqcomd 2448 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
(Scalar `  X )  =  (Scalar `  W )
)
65 eqidd 2444 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  (Scalar `  X
) )  =  (
Base `  (Scalar `  X
) ) )
661a1i 11 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  V  =  ( Base `  W ) )
675, 9ressplusg 14280 . . . . . 6  |-  ( U  e.  _V  ->  ( +g  `  W )  =  ( +g  `  X
) )
6861, 67syl 16 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( +g  `  W )  =  ( +g  `  X
) )
6968eqcomd 2448 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( +g  `  X )  =  ( +g  `  W
) )
705, 15ressvsca 14317 . . . . . 6  |-  ( U  e.  _V  ->  ( .s `  W )  =  ( .s `  X
) )
7161, 70syl 16 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( .s `  W
)  =  ( .s
`  X ) )
7271eqcomd 2448 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( .s `  X
)  =  ( .s
`  W ) )
732a1i 11 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  S  =  ( LSubSp `  W ) )
7459, 58eqsstr3d 3391 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  C_  V )
75 lmodgrp 16955 . . . . . 6  |-  ( X  e.  LMod  ->  X  e. 
Grp )
7675ad2antll 728 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  X  e.  Grp )
77 eqid 2443 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
7877grpbn0 15567 . . . . 5  |-  ( X  e.  Grp  ->  ( Base `  X )  =/=  (/) )
7976, 78syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  =/=  (/) )
80 eqid 2443 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
8177, 80lss1 17020 . . . . . 6  |-  ( X  e.  LMod  ->  ( Base `  X )  e.  (
LSubSp `  X ) )
8281ad2antll 728 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  ( LSubSp `  X )
)
83 eqid 2443 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
84 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
85 eqid 2443 . . . . . 6  |-  ( +g  `  X )  =  ( +g  `  X )
86 eqid 2443 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
8783, 84, 85, 86, 80lsscl 17024 . . . . 5  |-  ( ( ( Base `  X
)  e.  ( LSubSp `  X )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  a  e.  ( Base `  X )  /\  b  e.  ( Base `  X ) ) )  ->  ( (
x ( .s `  X ) a ) ( +g  `  X
) b )  e.  ( Base `  X
) )
8882, 87sylan 471 . . . 4  |-  ( ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  /\  (
x  e.  ( Base `  (Scalar `  X )
)  /\  a  e.  ( Base `  X )  /\  b  e.  ( Base `  X ) ) )  ->  ( (
x ( .s `  X ) a ) ( +g  `  X
) b )  e.  ( Base `  X
) )
8964, 65, 66, 69, 72, 73, 74, 79, 88islssd 17017 . . 3  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  S )
9059, 89eqeltrd 2517 . 2  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  S )
9157, 90impbida 828 1  |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  C_  V  /\  X  e. 
LMod ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972    C_ wss 3328   (/)c0 3637   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾s cress 14175   +g cplusg 14238   .rcmulr 14239  Scalarcsca 14241   .scvsca 14242   Grpcgrp 15410  SubGrpcsubg 15675   1rcur 16603   Ringcrg 16645   LModclmod 16948   LSubSpclss 17013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-sca 14254  df-vsca 14255  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lss 17014
This theorem is referenced by:  lsslmod  17041  lsslss  17042  issubassa  17395
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