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Theorem islss3 17718
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 17696 . . . 4  |-  ( U  e.  S  ->  U  C_  V )
43adantl 464 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  C_  V )
5 islss3.x . . . . . . 7  |-  X  =  ( Ws  U )
65, 1ressbas2 14692 . . . . . 6  |-  ( U 
C_  V  ->  U  =  ( Base `  X
) )
76adantl 464 . . . . 5  |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  =  ( Base `  X
) )
83, 7sylan2 472 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  =  ( Base `  X
) )
9 eqid 2382 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
105, 9ressplusg 14748 . . . . 5  |-  ( U  e.  S  ->  ( +g  `  W )  =  ( +g  `  X
) )
1110adantl 464 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  W )  =  ( +g  `  X
) )
12 eqid 2382 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
135, 12resssca 14784 . . . . 5  |-  ( U  e.  S  ->  (Scalar `  W )  =  (Scalar `  X ) )
1413adantl 464 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  =  (Scalar `  X ) )
15 eqid 2382 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
165, 15ressvsca 14785 . . . . 5  |-  ( U  e.  S  ->  ( .s `  W )  =  ( .s `  X
) )
1716adantl 464 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .s `  W )  =  ( .s `  X
) )
18 eqidd 2383 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  W
) ) )
19 eqidd 2383 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( +g  `  (Scalar `  W
) )  =  ( +g  `  (Scalar `  W ) ) )
20 eqidd 2383 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( .r `  (Scalar `  W
) )  =  ( .r `  (Scalar `  W ) ) )
21 eqidd 2383 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 1r `  (Scalar `  W
) )  =  ( 1r `  (Scalar `  W ) ) )
2212lmodring 17633 . . . . 5  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
2322adantr 463 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (Scalar `  W )  e.  Ring )
242lsssubg 17716 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
255subggrp 16321 . . . . 5  |-  ( U  e.  (SubGrp `  W
)  ->  X  e.  Grp )
2624, 25syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  Grp )
27 eqid 2382 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2812, 15, 27, 2lssvscl 17714 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U )
)  ->  ( x
( .s `  W
) a )  e.  U )
29283impb 1190 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  (
Base `  (Scalar `  W
) )  /\  a  e.  U )  ->  (
x ( .s `  W ) a )  e.  U )
30 simpll 751 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  W  e.  LMod )
31 simpr1 1000 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  x  e.  ( Base `  (Scalar `  W
) ) )
323ad2antlr 724 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  U  C_  V
)
33 simpr2 1001 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  U )
3432, 33sseldd 3418 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  a  e.  V )
35 simpr3 1002 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  U  /\  b  e.  U )
)  ->  b  e.  U )
3632, 35sseldd 3418 . . . . 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 17648 . . . . 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 1228 . . . 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 751 . . . . 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 1000 . . . . 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 1001 . . . . 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 724 . . . . . 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 1002 . . . . . 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 3418 . . . . 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 2382 . . . . . 6  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
461, 9, 12, 15, 27, 45lmodvsdir 17649 . . . . 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 1228 . . . 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 2382 . . . . . 6  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
491, 12, 15, 27, 48lmodvsass 17650 . . . . 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 1228 . . . 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 3417 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  U
)  ->  x  e.  V )
52 eqid 2382 . . . . . . 7  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
531, 12, 15, 52lmodvs1 17653 . . . . . 6  |-  ( ( W  e.  LMod  /\  x  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) x )  =  x )
5453adantlr 712 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  x  e.  V
)  ->  ( ( 1r `  (Scalar `  W
) ) ( .s
`  W ) x )  =  x )
5551, 54syldan 468 . . . 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 17631 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
574, 56jca 530 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( U  C_  V  /\  X  e.  LMod ) )
58 simprl 754 . . . 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 5784 . . . . . . 7  |-  ( Base `  X )  e.  _V
6159, 60syl6eqel 2478 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  _V )
625, 12resssca 14784 . . . . . 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 2390 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
(Scalar `  X )  =  (Scalar `  W )
)
65 eqidd 2383 . . . 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 14748 . . . . . 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 2390 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( +g  `  X )  =  ( +g  `  W
) )
705, 15ressvsca 14785 . . . . . 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 2390 . . . 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 3452 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  C_  V )
75 lmodgrp 17632 . . . . . 6  |-  ( X  e.  LMod  ->  X  e. 
Grp )
7675ad2antll 726 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  X  e.  Grp )
77 eqid 2382 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
7877grpbn0 16196 . . . . 5  |-  ( X  e.  Grp  ->  ( Base `  X )  =/=  (/) )
7976, 78syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  =/=  (/) )
80 eqid 2382 . . . . . . 7  |-  ( LSubSp `  X )  =  (
LSubSp `  X )
8177, 80lss1 17698 . . . . . 6  |-  ( X  e.  LMod  ->  ( Base `  X )  e.  (
LSubSp `  X ) )
8281ad2antll 726 . . . . 5  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  ( LSubSp `  X )
)
83 eqid 2382 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
84 eqid 2382 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
85 eqid 2382 . . . . . 6  |-  ( +g  `  X )  =  ( +g  `  X )
86 eqid 2382 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
8783, 84, 85, 86, 80lsscl 17702 . . . . 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 469 . . . 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 17695 . . 3  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  -> 
( Base `  X )  e.  S )
9059, 89eqeltrd 2470 . 2  |-  ( ( W  e.  LMod  /\  ( U  C_  V  /\  X  e.  LMod ) )  ->  U  e.  S )
9157, 90impbida 830 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   _Vcvv 3034    C_ wss 3389   (/)c0 3711   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   .rcmulr 14703  Scalarcsca 14705   .scvsca 14706   Grpcgrp 16170  SubGrpcsubg 16312   1rcur 17266   Ringcrg 17311   LModclmod 17625   LSubSpclss 17691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-sca 14718  df-vsca 14719  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-sbg 16176  df-subg 16315  df-mgp 17255  df-ur 17267  df-ring 17313  df-lmod 17627  df-lss 17692
This theorem is referenced by:  lsslmod  17719  lsslss  17720  issubassa  18086
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