Theorem lsatcvat3 29539

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23856 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 2408	. 2 |- ( <oLL ` W ) = ( <oLL ` W )
5		lsatcvat3.w	. 2 |- ( ph -> W e. LVec )
6		lveclmod 16137	. . . 4 |- ( W e. LVec -> W e. LMod )
7	5, 6	syl 16	. . 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 29484	. . . 4 |- ( ph -> Q e. S )
11		lsatcvat3.r	. . . . 5 |- ( ph -> R e. A )
12	1, 3, 7, 11	lsatlssel 29484	. . . 4 |- ( ph -> R e. S )
13	1, 2	lsmcl 16114	. . . 4 |- ( ( W e. LMod /\ Q e. S /\ R e. S ) -> ( Q .(+) R ) e. S )
14	7, 10, 12, 13	syl3anc 1184	. . 3 |- ( ph -> ( Q .(+) R ) e. S )
15	1	lssincl 16000	. . 3 |- ( ( W e. LMod /\ U e. S /\ ( Q .(+) R ) e. S ) -> ( U i^i ( Q .(+) R ) ) e. S )
16	7, 8, 14, 15	syl3anc 1184	. 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 29528	. . . . 5 |- ( ph -> ( -. R C_ U <-> U ( <oLL ` W ) ( U .(+) R ) ) )
20	18, 19	mpbid 202	. . . 4 |- ( ph -> U ( <oLL ` W ) ( U .(+) R ) )
21		lmodabl 15950	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
22	7, 21	syl 16	. . . . . . . . . 10 |- ( ph -> W e. Abel )
23	1	lsssssubg 15993	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
24	7, 23	syl 16	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
25	24, 10	sseldd 3313	. . . . . . . . . 10 |- ( ph -> Q e. (SubGrp ` W ) )
26	24, 12	sseldd 3313	. . . . . . . . . 10 |- ( ph -> R e. (SubGrp ` W ) )
27	2	lsmcom 15432	. . . . . . . . . 10 |- ( ( W e. Abel /\ Q e. (SubGrp ` W ) /\ R e. (SubGrp ` W ) ) -> ( Q .(+) R ) = ( R .(+) Q ) )
28	22, 25, 26, 27	syl3anc 1184	. . . . . . . . 9 |- ( ph -> ( Q .(+) R ) = ( R .(+) Q ) )
29	28	oveq2d 6060	. . . . . . . 8 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) ( R .(+) Q ) ) )
30	24, 8	sseldd 3313	. . . . . . . . 9 |- ( ph -> U e. (SubGrp ` W ) )
31	2	lsmass 15261	. . . . . . . . 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 1184	. . . . . . . 8 |- ( ph -> ( ( U .(+) R ) .(+) Q ) = ( U .(+) ( R .(+) Q ) ) )
33	29, 32	eqtr4d 2443	. . . . . . 7 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( ( U .(+) R ) .(+) Q ) )
34	1, 2	lsmcl 16114	. . . . . . . . . 10 |- ( ( W e. LMod /\ U e. S /\ R e. S ) -> ( U .(+) R ) e. S )
35	7, 8, 12, 34	syl3anc 1184	. . . . . . . . 9 |- ( ph -> ( U .(+) R ) e. S )
36	24, 35	sseldd 3313	. . . . . . . 8 |- ( ph -> ( U .(+) R ) e. (SubGrp ` W ) )
37		lsatcvat3.l	. . . . . . . 8 |- ( ph -> Q C_ ( U .(+) R ) )
38	2	lsmless2 15253	. . . . . . . 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 1184	. . . . . . 7 |- ( ph -> ( ( U .(+) R ) .(+) Q ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
40	33, 39	eqsstrd 3346	. . . . . 6 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
41	2	lsmidm 15255	. . . . . . 7 |- ( ( U .(+) R ) e. (SubGrp ` W ) -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) )
42	36, 41	syl 16	. . . . . 6 |- ( ph -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) )
43	40, 42	sseqtrd 3348	. . . . 5 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( U .(+) R ) )
44	24, 14	sseldd 3313	. . . . . 6 |- ( ph -> ( Q .(+) R ) e. (SubGrp ` W ) )
45	2	lsmub2 15250	. . . . . . 7 |- ( ( Q e. (SubGrp ` W ) /\ R e. (SubGrp ` W ) ) -> R C_ ( Q .(+) R ) )
46	25, 26, 45	syl2anc 643	. . . . . 6 |- ( ph -> R C_ ( Q .(+) R ) )
47	2	lsmless2 15253	. . . . . 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 1184	. . . . 5 |- ( ph -> ( U .(+) R ) C_ ( U .(+) ( Q .(+) R ) ) )
49	43, 48	eqssd 3329	. . . 4 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) R ) )
50	20, 49	breqtrrd 4202	. . 3 |- ( ph -> U ( <oLL ` W ) ( U .(+) ( Q .(+) R ) ) )
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 29524	. 2 |- ( ph -> ( U i^i ( Q .(+) R ) ) ( <oLL ` W ) ( Q .(+) R ) )
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 29538	1 |- ( ph -> ( U i^i ( Q .(+) R ) ) e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1649   e. wcel 1721   =/= wne 2571   i^i cin 3283   C_ wss 3284   class class class wbr 4176   ` cfv 5417  (class class class)co 6044  SubGrpcsubg 14897   LSSumclsm 15227   Abelcabel 15372   LModclmod 15909   LSubSpclss 15967   LVecclvec 16133  LSAtomsclsa 29461   <oL_L clcv 29505

This theorem is referenced by:  l1cvat  29542

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-tpos 6442  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-0g 13686  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-oppg 15101  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lcv 29506