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Theorem islss3 15765
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 15743 . . . 4  |-  ( U  e.  S  ->  U  C_  V )
43adantl 452 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  C_  V )
5 islss3.x . . . . . . 7  |-  X  =  ( Ws  U )
65, 1ressbas2 13246 . . . . . 6  |-  ( U 
C_  V  ->  U  =  ( Base `  X
) )
76adantl 452 . . . . 5  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  =  ( Base `  X
) )
83, 7sylan2 460 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  =  ( Base `  X
) )
9 eqid 2316 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
105, 9ressplusg 13297 . . . . 5  |-  ( U  e.  S  ->  ( +g  `  W )  =  ( +g  `  X
) )
1110adantl 452 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  W )  =  ( +g  `  X
) )
12 eqid 2316 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
135, 12resssca 13330 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 452 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
15 eqid 2316 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
165, 15ressvsca 13331 . . . . 5  |-  ( U  e.  S  ->  ( .s `  W )  =  ( .s `  X
) )
1716adantl 452 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .s `  W )  =  ( .s `  X
) )
18 eqidd 2317 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
19 eqidd 2317 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  (Scalar `  W
) )  =  ( +g  `  (Scalar `  W ) ) )
20 eqidd 2317 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .r `  (Scalar `  W
) )  =  ( .r `  (Scalar `  W ) ) )
21 eqidd 2317 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 1r `  (Scalar `  W
) )  =  ( 1r `  (Scalar `  W ) ) )
2212lmodrng 15684 . . . . 5  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
2322adantr 451 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  e.  Ring )
242lsssubg 15763 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
255subggrp 14673 . . . . 5  |-  ( U  e.  (SubGrp `  W
)  ->  X  e.  Grp )
2624, 25syl 15 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  Grp )
27 eqid 2316 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2812, 15, 27, 2lssvscl 15761 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U )
)  ->  ( x
( .s `  W
) a )  e.  U )
29283impb 1147 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  (
Base `  (Scalar `  W
) )  /\  a  e.  U )  ->  (
x ( .s `  W ) a )  e.  U )
30 simpll 730 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  W  e.  LMod )
31 simpr1 961 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  x  e.  ( Base `  (Scalar `  W
) ) )
323ad2antlr 707 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  U  C_  V
)
33 simpr2 962 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  U )
3432, 33sseldd 3215 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  V )
35 simpr3 963 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  b  e.  U )
3632, 35sseldd 3215 . . . . 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 15699 . . . . 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 1184 . . . 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 730 . . . . 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 961 . . . . 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 962 . . . . 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 707 . . . . . 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 963 . . . . . 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 3215 . . . . 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 2316 . . . . . 6  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
461, 9, 12, 15, 27, 45lmodvsdir 15701 . . . . 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 1184 . . . 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 2316 . . . . . 6  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
491, 12, 15, 27, 48lmodvsass 15703 . . . . 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 1184 . . . 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 3214 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  U
)  ->  x  e.  V )
52 eqid 2316 . . . . . . 7  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
531, 12, 15, 52lmodvs1 15707 . . . . . 6  |-  ( ( W  e.  LMod  /\  x  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) x )  =  x )
5453adantlr 695 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  V
)  ->  ( ( 1r `  (Scalar `  W
) ) ( .s
`  W ) x )  =  x )
5551, 54syldan 456 . . . 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 15682 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
574, 56jca 518 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  V  /\  X  e.  LMod ) )
58 simprl 732 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  C_  V )
5958, 6syl 15 . . 3  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  =  ( Base `  X ) )
60 fvex 5577 . . . . . . 7  |-  ( Base `  X )  e.  _V
6159, 60syl6eqel 2404 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  _V )
625, 12resssca 13330 . . . . . 6  |-  ( U  e.  _V  ->  (Scalar `  W )  =  (Scalar `  X ) )
6361, 62syl 15 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
(Scalar `  W )  =  (Scalar `  X )
)
6463eqcomd 2321 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
(Scalar `  X )  =  (Scalar `  W )
)
65 eqidd 2317 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  (Scalar `  X
) )  =  (
Base `  (Scalar `  X
) ) )
661a1i 10 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  V  =  ( Base `  W ) )
675, 9ressplusg 13297 . . . . . 6  |-  ( U  e.  _V  ->  ( +g  `  W )  =  ( +g  `  X
) )
6861, 67syl 15 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( +g  `  W )  =  ( +g  `  X
) )
6968eqcomd 2321 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( +g  `  X )  =  ( +g  `  W
) )
705, 15ressvsca 13331 . . . . . 6  |-  ( U  e.  _V  ->  ( .s `  W )  =  ( .s `  X
) )
7161, 70syl 15 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( .s `  W
)  =  ( .s
`  X ) )
7271eqcomd 2321 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( .s `  X
)  =  ( .s
`  W ) )
732a1i 10 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  S  =  ( LSubSp `  W ) )
7459, 58eqsstr3d 3247 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  C_  V )
75 lmodgrp 15683 . . . . . 6  |-  ( X  e.  LMod  ->  X  e. 
Grp )
7675ad2antll 709 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  X  e.  Grp )
77 eqid 2316 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
7877grpbn0 14560 . . . . 5  |-  ( X  e.  Grp  ->  ( Base `  X )  =/=  (/) )
7976, 78syl 15 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  =/=  (/) )
80 eqid 2316 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
8177, 80lss1 15745 . . . . . 6  |-  ( X  e.  LMod  ->  ( Base `  X )  e.  (
LSubSp `  X ) )
8281ad2antll 709 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  ( LSubSp `  X )
)
83 eqid 2316 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
84 eqid 2316 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
85 eqid 2316 . . . . . 6  |-  ( +g  `  X )  =  ( +g  `  X )
86 eqid 2316 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
8783, 84, 85, 86, 80lsscl 15749 . . . . 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 457 . . . 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 15742 . . 3  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  S )
9059, 89eqeltrd 2390 . 2  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  S )
9157, 90impbida 805 1  |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  C_  V  /\  X  e. 
LMod ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   _Vcvv 2822    C_ wss 3186   (/)c0 3489   ` cfv 5292  (class class class)co 5900   Basecbs 13195   ↾s cress 13196   +g cplusg 13255   .rcmulr 13256  Scalarcsca 13258   .scvsca 13259   Grpcgrp 14411  SubGrpcsubg 14664   Ringcrg 15386   1rcur 15388   LModclmod 15676   LSubSpclss 15738
This theorem is referenced by:  lsslmod  15766  lsslss  15767  issubassa  16113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-sca 13271  df-vsca 13272  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-mgp 15375  df-rng 15389  df-ur 15391  df-lmod 15678  df-lss 15739
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