Theorem lsatcvat3 28521

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22972 analog.) (Contributed by NM, 11-Jan-2015.)

Hypotheses

Ref	Expression
lsatcvat3.s	|- S = ( LSubSp ` W )
lsatcvat3.p	|- .(+) = ( LSSum ` W )
lsatcvat3.a	|- A = (LSAtoms ` W )
lsatcvat3.w	|- ( ph -> W e. LVec )
lsatcvat3.u	|- ( ph -> U e. S )
lsatcvat3.q	|- ( ph -> Q e. A )
lsatcvat3.r	|- ( ph -> R e. A )
lsatcvat3.n	|- ( ph -> Q =/= R )
lsatcvat3.m	|- ( ph -> -. R C_ U )
lsatcvat3.l	|- ( ph -> Q C_ ( U .(+) R ) )

Assertion

Ref	Expression
lsatcvat3	|- ( ph -> ( U i^i ( Q .(+) R ) ) e. A )

Proof of Theorem lsatcvat3

Step	Hyp	Ref	Expression
1		lsatcvat3.s	. 2 |- S = ( LSubSp ` W )
2		lsatcvat3.p	. 2 |- .(+) = ( LSSum ` W )
3		lsatcvat3.a	. 2 |- A = (LSAtoms ` W )
4		eqid 2284	. 2 |- ( <oLL ` W ) = ( <oLL ` W )
5		lsatcvat3.w	. 2 |- ( ph -> W e. LVec )
6		lveclmod 15855	. . . 4 |- ( W e. LVec -> W e. LMod )
7	5, 6	syl 15	. . 3 |- ( ph -> W e. LMod )
8		lsatcvat3.u	. . 3 |- ( ph -> U e. S )
9		lsatcvat3.q	. . . . 5 |- ( ph -> Q e. A )
10	1, 3, 7, 9	lsatlssel 28466	. . . 4 |- ( ph -> Q e. S )
11		lsatcvat3.r	. . . . 5 |- ( ph -> R e. A )
12	1, 3, 7, 11	lsatlssel 28466	. . . 4 |- ( ph -> R e. S )
13	1, 2	lsmcl 15832	. . . 4 |- ( ( W e. LMod /\ Q e. S /\ R e. S ) -> ( Q .(+) R ) e. S )
14	7, 10, 12, 13	syl3anc 1182	. . 3 |- ( ph -> ( Q .(+) R ) e. S )
15	1	lssincl 15718	. . 3 |- ( ( W e. LMod /\ U e. S /\ ( Q .(+) R ) e. S ) -> ( U i^i ( Q .(+) R ) ) e. S )
16	7, 8, 14, 15	syl3anc 1182	. 2 |- ( ph -> ( U i^i ( Q .(+) R ) ) e. S )
17		lsatcvat3.n	. 2 |- ( ph -> Q =/= R )
18		lsatcvat3.m	. . . . 5 |- ( ph -> -. R C_ U )
19	1, 2, 3, 4, 5, 8, 11	lcv1 28510	. . . . 5 |- ( ph -> ( -. R C_ U <-> U ( <oLL ` W ) ( U .(+) R ) ) )
20	18, 19	mpbid 201	. . . 4 |- ( ph -> U ( <oLL ` W ) ( U .(+) R ) )
21		lmodabl 15668	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
22	7, 21	syl 15	. . . . . . . . . 10 |- ( ph -> W e. Abel )
23	1	lsssssubg 15711	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
24	7, 23	syl 15	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
25	24, 10	sseldd 3182	. . . . . . . . . 10 |- ( ph -> Q e. (SubGrp ` W ) )
26	24, 12	sseldd 3182	. . . . . . . . . 10 |- ( ph -> R e. (SubGrp ` W ) )
27	2	lsmcom 15146	. . . . . . . . . 10 |- ( ( W e. Abel /\ Q e. (SubGrp ` W ) /\ R e. (SubGrp ` W ) ) -> ( Q .(+) R ) = ( R .(+) Q ) )
28	22, 25, 26, 27	syl3anc 1182	. . . . . . . . 9 |- ( ph -> ( Q .(+) R ) = ( R .(+) Q ) )
29	28	oveq2d 5836	. . . . . . . 8 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) ( R .(+) Q ) ) )
30	24, 8	sseldd 3182	. . . . . . . . 9 |- ( ph -> U e. (SubGrp ` W ) )
31	2	lsmass 14975	. . . . . . . . 9 |- ( ( U e. (SubGrp ` W ) /\ R e. (SubGrp ` W ) /\ Q e. (SubGrp ` W ) ) -> ( ( U .(+) R ) .(+) Q ) = ( U .(+) ( R .(+) Q ) ) )
32	30, 26, 25, 31	syl3anc 1182	. . . . . . . 8 |- ( ph -> ( ( U .(+) R ) .(+) Q ) = ( U .(+) ( R .(+) Q ) ) )
33	29, 32	eqtr4d 2319	. . . . . . 7 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( ( U .(+) R ) .(+) Q ) )
34	1, 2	lsmcl 15832	. . . . . . . . . 10 |- ( ( W e. LMod /\ U e. S /\ R e. S ) -> ( U .(+) R ) e. S )
35	7, 8, 12, 34	syl3anc 1182	. . . . . . . . 9 |- ( ph -> ( U .(+) R ) e. S )
36	24, 35	sseldd 3182	. . . . . . . 8 |- ( ph -> ( U .(+) R ) e. (SubGrp ` W ) )
37		lsatcvat3.l	. . . . . . . 8 |- ( ph -> Q C_ ( U .(+) R ) )
38	2	lsmless2 14967	. . . . . . . 8 |- ( ( ( U .(+) R ) e. (SubGrp ` W ) /\ ( U .(+) R ) e. (SubGrp ` W ) /\ Q C_ ( U .(+) R ) ) -> ( ( U .(+) R ) .(+) Q ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
39	36, 36, 37, 38	syl3anc 1182	. . . . . . 7 |- ( ph -> ( ( U .(+) R ) .(+) Q ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
40	33, 39	eqsstrd 3213	. . . . . 6 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
41	2	lsmidm 14969	. . . . . . 7 |- ( ( U .(+) R ) e. (SubGrp ` W ) -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) )
42	36, 41	syl 15	. . . . . 6 |- ( ph -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) )
43	40, 42	sseqtrd 3215	. . . . 5 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( U .(+) R ) )
44	24, 14	sseldd 3182	. . . . . 6 |- ( ph -> ( Q .(+) R ) e. (SubGrp ` W ) )
45	2	lsmub2 14964	. . . . . . 7 |- ( ( Q e. (SubGrp ` W ) /\ R e. (SubGrp ` W ) ) -> R C_ ( Q .(+) R ) )
46	25, 26, 45	syl2anc 642	. . . . . 6 |- ( ph -> R C_ ( Q .(+) R ) )
47	2	lsmless2 14967	. . . . . 6 |- ( ( U e. (SubGrp ` W ) /\ ( Q .(+) R ) e. (SubGrp ` W ) /\ R C_ ( Q .(+) R ) ) -> ( U .(+) R ) C_ ( U .(+) ( Q .(+) R ) ) )
48	30, 44, 46, 47	syl3anc 1182	. . . . 5 |- ( ph -> ( U .(+) R ) C_ ( U .(+) ( Q .(+) R ) ) )
49	43, 48	eqssd 3197	. . . 4 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) R ) )
50	20, 49	breqtrrd 4050	. . 3 |- ( ph -> U ( <oLL ` W ) ( U .(+) ( Q .(+) R ) ) )
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 28506	. 2 |- ( ph -> ( U i^i ( Q .(+) R ) ) ( <oLL ` W ) ( Q .(+) R ) )
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 28520	1 |- ( ph -> ( U i^i ( Q .(+) R ) ) e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1623   e. wcel 1685   =/= wne 2447   i^i cin 3152   C_ wss 3153   class class class wbr 4024   ` cfv 5221  (class class class)co 5820  SubGrpcsubg 14611   LSSumclsm 14941   Abelcabel 15086   LModclmod 15623   LSubSpclss 15685   LVecclvec 15851  LSAtomsclsa 28443   <oL_L clcv 28487

This theorem is referenced by:  l1cvat  28524

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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810

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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-0g 13400  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-grp 14485  df-minusg 14486  df-sbg 14487  df-subg 14614  df-cntz 14789  df-oppg 14815  df-lsm 14943  df-cmn 15087  df-abl 15088  df-mgp 15322  df-rng 15336  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-invr 15450  df-drng 15510  df-lmod 15625  df-lss 15686  df-lsp 15725  df-lvec 15852  df-lsatoms 28445  df-lcv 28488