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Theorem islss3 16035
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 16013 . . . 4  |-  ( U  e.  S  ->  U  C_  V )
43adantl 453 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  C_  V )
5 islss3.x . . . . . . 7  |-  X  =  ( Ws  U )
65, 1ressbas2 13520 . . . . . 6  |-  ( U 
C_  V  ->  U  =  ( Base `  X
) )
76adantl 453 . . . . 5  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  =  ( Base `  X
) )
83, 7sylan2 461 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  =  ( Base `  X
) )
9 eqid 2436 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
105, 9ressplusg 13571 . . . . 5  |-  ( U  e.  S  ->  ( +g  `  W )  =  ( +g  `  X
) )
1110adantl 453 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  W )  =  ( +g  `  X
) )
12 eqid 2436 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
135, 12resssca 13604 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 453 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
15 eqid 2436 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
165, 15ressvsca 13605 . . . . 5  |-  ( U  e.  S  ->  ( .s `  W )  =  ( .s `  X
) )
1716adantl 453 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .s `  W )  =  ( .s `  X
) )
18 eqidd 2437 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
19 eqidd 2437 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  (Scalar `  W
) )  =  ( +g  `  (Scalar `  W ) ) )
20 eqidd 2437 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .r `  (Scalar `  W
) )  =  ( .r `  (Scalar `  W ) ) )
21 eqidd 2437 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 1r `  (Scalar `  W
) )  =  ( 1r `  (Scalar `  W ) ) )
2212lmodrng 15958 . . . . 5  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
2322adantr 452 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  e.  Ring )
242lsssubg 16033 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
255subggrp 14947 . . . . 5  |-  ( U  e.  (SubGrp `  W
)  ->  X  e.  Grp )
2624, 25syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  Grp )
27 eqid 2436 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2812, 15, 27, 2lssvscl 16031 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U )
)  ->  ( x
( .s `  W
) a )  e.  U )
29283impb 1149 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  (
Base `  (Scalar `  W
) )  /\  a  e.  U )  ->  (
x ( .s `  W ) a )  e.  U )
30 simpll 731 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  W  e.  LMod )
31 simpr1 963 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  x  e.  ( Base `  (Scalar `  W
) ) )
323ad2antlr 708 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  U  C_  V
)
33 simpr2 964 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  U )
3432, 33sseldd 3349 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  V )
35 simpr3 965 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  b  e.  U )
3632, 35sseldd 3349 . . . . 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 15973 . . . . 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 1186 . . . 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 731 . . . . 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 963 . . . . 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 964 . . . . 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 708 . . . . . 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 965 . . . . . 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 3349 . . . . 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 2436 . . . . . 6  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
461, 9, 12, 15, 27, 45lmodvsdir 15974 . . . . 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 1186 . . . 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 2436 . . . . . 6  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
491, 12, 15, 27, 48lmodvsass 15975 . . . . 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 1186 . . . 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 3348 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  U
)  ->  x  e.  V )
52 eqid 2436 . . . . . . 7  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
531, 12, 15, 52lmodvs1 15978 . . . . . 6  |-  ( ( W  e.  LMod  /\  x  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) x )  =  x )
5453adantlr 696 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  V
)  ->  ( ( 1r `  (Scalar `  W
) ) ( .s
`  W ) x )  =  x )
5551, 54syldan 457 . . . 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 15956 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
574, 56jca 519 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  V  /\  X  e.  LMod ) )
58 simprl 733 . . . 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 5742 . . . . . . 7  |-  ( Base `  X )  e.  _V
6159, 60syl6eqel 2524 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  _V )
625, 12resssca 13604 . . . . . 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 2441 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
(Scalar `  X )  =  (Scalar `  W )
)
65 eqidd 2437 . . . 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 13571 . . . . . 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 2441 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( +g  `  X )  =  ( +g  `  W
) )
705, 15ressvsca 13605 . . . . . 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 2441 . . . 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 3383 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  C_  V )
75 lmodgrp 15957 . . . . . 6  |-  ( X  e.  LMod  ->  X  e. 
Grp )
7675ad2antll 710 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  X  e.  Grp )
77 eqid 2436 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
7877grpbn0 14834 . . . . 5  |-  ( X  e.  Grp  ->  ( Base `  X )  =/=  (/) )
7976, 78syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  =/=  (/) )
80 eqid 2436 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
8177, 80lss1 16015 . . . . . 6  |-  ( X  e.  LMod  ->  ( Base `  X )  e.  (
LSubSp `  X ) )
8281ad2antll 710 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  ( LSubSp `  X )
)
83 eqid 2436 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
84 eqid 2436 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
85 eqid 2436 . . . . . 6  |-  ( +g  `  X )  =  ( +g  `  X )
86 eqid 2436 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
8783, 84, 85, 86, 80lsscl 16019 . . . . 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 458 . . . 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 16012 . . 3  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  S )
9059, 89eqeltrd 2510 . 2  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  S )
9157, 90impbida 806 1  |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  C_  V  /\  X  e. 
LMod ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   Grpcgrp 14685  SubGrpcsubg 14938   Ringcrg 15660   1rcur 15662   LModclmod 15950   LSubSpclss 16008
This theorem is referenced by:  lsslmod  16036  lsslss  16037  issubassa  16383
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-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-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  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-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009
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