Theorem lsslss 14655

Description: The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)

Hypotheses

Ref	Expression
lsslss.x	|- X = ( W ↾s U )
lsslss.s	|- S = ( LSubSp ` W )
lsslss.t	|- T = ( LSubSp ` X )

Assertion

Ref	Expression
lsslss	|- ( ( W e. LMod /\ U e. S ) -> ( V e. T <-> ( V e. S /\ V C_ U ) ) )

Proof of Theorem lsslss

Step	Hyp	Ref	Expression
1		lsslss.x	. . . 4 |- X = ( W ↾s U )
2		lsslss.s	. . . 4 |- S = ( LSubSp ` W )
3	1, 2	lsslmod 14654	. . 3 |- ( ( W e. LMod /\ U e. S ) -> X e. LMod )
4		eqid 2234	. . . 4 |- ( X ↾s V ) = ( X ↾s V )
5		eqid 2234	. . . 4 |- ( Base ` X ) = ( Base ` X )
6		lsslss.t	. . . 4 |- T = ( LSubSp ` X )
7	4, 5, 6	islss3 14653	. . 3 |- ( X e. LMod -> ( V e. T <-> ( V C_ ( Base ` X ) /\ ( X ↾s V ) e. LMod ) ) )
8	3, 7	syl 14	. 2 |- ( ( W e. LMod /\ U e. S ) -> ( V e. T <-> ( V C_ ( Base ` X ) /\ ( X ↾s V ) e. LMod ) ) )
9	1	a1i 9	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> X = ( W ↾s U ) )
10		eqid 2234	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
11	10	a1i 9	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> ( Base ` W ) = ( Base ` W ) )
12		simpl 109	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> W e. LMod )
13	10, 2	lssssg 14634	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U C_ ( Base ` W ) )
14	9, 11, 12, 13	ressbas2d 13365	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> U = ( Base ` X ) )
15	14	sseq2d 3272	. . 3 |- ( ( W e. LMod /\ U e. S ) -> ( V C_ U <-> V C_ ( Base ` X ) ) )
16	15	anbi1d 465	. 2 |- ( ( W e. LMod /\ U e. S ) -> ( ( V C_ U /\ ( X ↾s V ) e. LMod ) <-> ( V C_ ( Base ` X ) /\ ( X ↾s V ) e. LMod ) ) )
17		sstr2 3249	. . . . . . 7 |- ( V C_ U -> ( U C_ ( Base ` W ) -> V C_ ( Base ` W ) ) )
18	13, 17	mpan9 281	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> V C_ ( Base ` W ) )
19	18	biantrurd 305	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> ( ( W ↾s V ) e. LMod <-> ( V C_ ( Base ` W ) /\ ( W ↾s V ) e. LMod ) ) )
20	1	oveq1i 6068	. . . . . . 7 |- ( X ↾s V ) = ( ( W ↾s U ) ↾s V )
21		simplr 529	. . . . . . . 8 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> U e. S )
22		simpr 110	. . . . . . . 8 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> V C_ U )
23		simpll 527	. . . . . . . 8 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> W e. LMod )
24		ressabsg 13373	. . . . . . . 8 |- ( ( U e. S /\ V C_ U /\ W e. LMod ) -> ( ( W ↾s U ) ↾s V ) = ( W ↾s V ) )
25	21, 22, 23, 24	syl3anc 1274	. . . . . . 7 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> ( ( W ↾s U ) ↾s V ) = ( W ↾s V ) )
26	20, 25	eqtrid 2279	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> ( X ↾s V ) = ( W ↾s V ) )
27	26	eleq1d 2303	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> ( ( X ↾s V ) e. LMod <-> ( W ↾s V ) e. LMod ) )
28		eqid 2234	. . . . . . 7 |- ( W ↾s V ) = ( W ↾s V )
29	28, 10, 2	islss3 14653	. . . . . 6 |- ( W e. LMod -> ( V e. S <-> ( V C_ ( Base ` W ) /\ ( W ↾s V ) e. LMod ) ) )
30	29	ad2antrr 488	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> ( V e. S <-> ( V C_ ( Base ` W ) /\ ( W ↾s V ) e. LMod ) ) )
31	19, 27, 30	3bitr4d 220	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ V C_ U ) -> ( ( X ↾s V ) e. LMod <-> V e. S ) )
32	31	pm5.32da 452	. . 3 |- ( ( W e. LMod /\ U e. S ) -> ( ( V C_ U /\ ( X ↾s V ) e. LMod ) <-> ( V C_ U /\ V e. S ) ) )
33	32	biancomd 271	. 2 |- ( ( W e. LMod /\ U e. S ) -> ( ( V C_ U /\ ( X ↾s V ) e. LMod ) <-> ( V e. S /\ V C_ U ) ) )
34	8, 16, 33	3bitr2d 216	1 |- ( ( W e. LMod /\ U e. S ) -> ( V e. T <-> ( V e. S /\ V C_ U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   C_ wss 3214   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   LModclmod 14561   LSubSpclss 14626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-sbg 13760  df-subg 13923  df-mgp 14160  df-ur 14203  df-ring 14241  df-lmod 14563  df-lssm 14627

This theorem is referenced by:  lsslsp  14703