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Theorem islss3 15716
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 15694 . . . 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 13199 . . . . . 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 2283 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
105, 9ressplusg 13250 . . . . 5  |-  ( U  e.  S  ->  ( +g  `  W )  =  ( +g  `  X
) )
1110adantl 452 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  W )  =  ( +g  `  X
) )
12 eqid 2283 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
135, 12resssca 13283 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 452 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
15 eqid 2283 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
165, 15ressvsca 13284 . . . . 5  |-  ( U  e.  S  ->  ( .s `  W )  =  ( .s `  X
) )
1716adantl 452 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .s `  W )  =  ( .s `  X
) )
18 eqidd 2284 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
19 eqidd 2284 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  (Scalar `  W
) )  =  ( +g  `  (Scalar `  W ) ) )
20 eqidd 2284 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .r `  (Scalar `  W
) )  =  ( .r `  (Scalar `  W ) ) )
21 eqidd 2284 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 1r `  (Scalar `  W
) )  =  ( 1r `  (Scalar `  W ) ) )
2212lmodrng 15635 . . . . 5  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
2322adantr 451 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  e.  Ring )
242lsssubg 15714 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
255subggrp 14624 . . . . 5  |-  ( U  e.  (SubGrp `  W
)  ->  X  e.  Grp )
2624, 25syl 15 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  Grp )
27 eqid 2283 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2812, 15, 27, 2lssvscl 15712 . . . . 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 3181 . . . . 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 3181 . . . . 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 15650 . . . . 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 3181 . . . . 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 2283 . . . . . 6  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
461, 9, 12, 15, 27, 45lmodvsdir 15652 . . . . 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 2283 . . . . . 6  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
491, 12, 15, 27, 48lmodvsass 15654 . . . . 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 3180 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  U
)  ->  x  e.  V )
52 eqid 2283 . . . . . . 7  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
531, 12, 15, 52lmodvs1 15658 . . . . . 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 15633 . . 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 5539 . . . . . . 7  |-  ( Base `  X )  e.  _V
6159, 60syl6eqel 2371 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  _V )
625, 12resssca 13283 . . . . . 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 2288 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
(Scalar `  X )  =  (Scalar `  W )
)
65 eqidd 2284 . . . 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 13250 . . . . . 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 2288 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( +g  `  X )  =  ( +g  `  W
) )
705, 15ressvsca 13284 . . . . . 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 2288 . . . 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 3213 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  C_  V )
75 lmodgrp 15634 . . . . . 6  |-  ( X  e.  LMod  ->  X  e. 
Grp )
7675ad2antll 709 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  X  e.  Grp )
77 eqid 2283 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
7877grpbn0 14511 . . . . 5  |-  ( X  e.  Grp  ->  ( Base `  X )  =/=  (/) )
7976, 78syl 15 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  =/=  (/) )
80 eqid 2283 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
8177, 80lss1 15696 . . . . . 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 2283 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
84 eqid 2283 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
85 eqid 2283 . . . . . 6  |-  ( +g  `  X )  =  ( +g  `  X )
86 eqid 2283 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
8783, 84, 85, 86, 80lsscl 15700 . . . . 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 15693 . . 3  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  S )
9059, 89eqeltrd 2357 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   Grpcgrp 14362  SubGrpcsubg 14615   Ringcrg 15337   1rcur 15339   LModclmod 15627   LSubSpclss 15689
This theorem is referenced by:  lsslmod  15717  lsslss  15718  issubassa  16064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690
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