Theorem lsatcvat3 29787

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23891 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 2435	. 2 |- ( <oLL ` W ) = ( <oLL ` W )
5		lsatcvat3.w	. 2 |- ( ph -> W e. LVec )
6		lveclmod 16170	. . . 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 29732	. . . 4 |- ( ph -> Q e. S )
11		lsatcvat3.r	. . . . 5 |- ( ph -> R e. A )
12	1, 3, 7, 11	lsatlssel 29732	. . . 4 |- ( ph -> R e. S )
13	1, 2	lsmcl 16147	. . . 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 16033	. . 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 29776	. . . . 5 |- ( ph -> ( -. R C_ U <-> U ( <oLL ` W ) ( U .(+) R ) ) )
20	18, 19	mpbid 202	. . . 4 |- ( ph -> U ( <oLL ` W ) ( U .(+) R ) )
21		lmodabl 15983	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
22	7, 21	syl 16	. . . . . . . . . 10 |- ( ph -> W e. Abel )
23	1	lsssssubg 16026	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
24	7, 23	syl 16	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
25	24, 10	sseldd 3341	. . . . . . . . . 10 |- ( ph -> Q e. (SubGrp ` W ) )
26	24, 12	sseldd 3341	. . . . . . . . . 10 |- ( ph -> R e. (SubGrp ` W ) )
27	2	lsmcom 15465	. . . . . . . . . 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 6089	. . . . . . . 8 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) ( R .(+) Q ) ) )
30	24, 8	sseldd 3341	. . . . . . . . 9 |- ( ph -> U e. (SubGrp ` W ) )
31	2	lsmass 15294	. . . . . . . . 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 2470	. . . . . . 7 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( ( U .(+) R ) .(+) Q ) )
34	1, 2	lsmcl 16147	. . . . . . . . . 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 3341	. . . . . . . 8 |- ( ph -> ( U .(+) R ) e. (SubGrp ` W ) )
37		lsatcvat3.l	. . . . . . . 8 |- ( ph -> Q C_ ( U .(+) R ) )
38	2	lsmless2 15286	. . . . . . . 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 3374	. . . . . 6 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
41	2	lsmidm 15288	. . . . . . 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 3376	. . . . 5 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( U .(+) R ) )
44	24, 14	sseldd 3341	. . . . . 6 |- ( ph -> ( Q .(+) R ) e. (SubGrp ` W ) )
45	2	lsmub2 15283	. . . . . . 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 15286	. . . . . 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 3357	. . . 4 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) R ) )
50	20, 49	breqtrrd 4230	. . 3 |- ( ph -> U ( <oLL ` W ) ( U .(+) ( Q .(+) R ) ) )
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 29772	. 2 |- ( ph -> ( U i^i ( Q .(+) R ) ) ( <oLL ` W ) ( Q .(+) R ) )
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 29786	1 |- ( ph -> ( U i^i ( Q .(+) R ) ) e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1652   e. wcel 1725   =/= wne 2598   i^i cin 3311   C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073  SubGrpcsubg 14930   LSSumclsm 15260   Abelcabel 15405   LModclmod 15942   LSubSpclss 16000   LVecclvec 16166  LSAtomsclsa 29709   <oL_L clcv 29753

This theorem is referenced by:  l1cvat  29790

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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059

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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29711  df-lcv 29754