Theorem lsatcvat3 28492

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22937 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 2258	. 2 |- ( <oLL ` W ) = ( <oLL ` W )
5		lsatcvat3.w	. 2 |- ( ph -> W e. LVec )
6		lveclmod 15822	. . . 4 |- ( W e. LVec -> W e. LMod )
7	5, 6	syl 17	. . 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 28437	. . . 4 |- ( ph -> Q e. S )
11		lsatcvat3.r	. . . . 5 |- ( ph -> R e. A )
12	1, 3, 7, 11	lsatlssel 28437	. . . 4 |- ( ph -> R e. S )
13	1, 2	lsmcl 15799	. . . 4 |- ( ( W e. LMod /\ Q e. S /\ R e. S ) -> ( Q .(+) R ) e. S )
14	7, 10, 12, 13	syl3anc 1187	. . 3 |- ( ph -> ( Q .(+) R ) e. S )
15	1	lssincl 15685	. . 3 |- ( ( W e. LMod /\ U e. S /\ ( Q .(+) R ) e. S ) -> ( U i^i ( Q .(+) R ) ) e. S )
16	7, 8, 14, 15	syl3anc 1187	. 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 28481	. . . . 5 |- ( ph -> ( -. R C_ U <-> U ( <oLL ` W ) ( U .(+) R ) ) )
20	18, 19	mpbid 203	. . . 4 |- ( ph -> U ( <oLL ` W ) ( U .(+) R ) )
21		lmodabl 15635	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
22	7, 21	syl 17	. . . . . . . . . 10 |- ( ph -> W e. Abel )
23	1	lsssssubg 15678	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
24	7, 23	syl 17	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
25	24, 10	sseldd 3156	. . . . . . . . . 10 |- ( ph -> Q e. (SubGrp ` W ) )
26	24, 12	sseldd 3156	. . . . . . . . . 10 |- ( ph -> R e. (SubGrp ` W ) )
27	2	lsmcom 15113	. . . . . . . . . 10 |- ( ( W e. Abel /\ Q e. (SubGrp ` W ) /\ R e. (SubGrp ` W ) ) -> ( Q .(+) R ) = ( R .(+) Q ) )
28	22, 25, 26, 27	syl3anc 1187	. . . . . . . . 9 |- ( ph -> ( Q .(+) R ) = ( R .(+) Q ) )
29	28	oveq2d 5808	. . . . . . . 8 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) ( R .(+) Q ) ) )
30	24, 8	sseldd 3156	. . . . . . . . 9 |- ( ph -> U e. (SubGrp ` W ) )
31	2	lsmass 14942	. . . . . . . . 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 1187	. . . . . . . 8 |- ( ph -> ( ( U .(+) R ) .(+) Q ) = ( U .(+) ( R .(+) Q ) ) )
33	29, 32	eqtr4d 2293	. . . . . . 7 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( ( U .(+) R ) .(+) Q ) )
34	1, 2	lsmcl 15799	. . . . . . . . . 10 |- ( ( W e. LMod /\ U e. S /\ R e. S ) -> ( U .(+) R ) e. S )
35	7, 8, 12, 34	syl3anc 1187	. . . . . . . . 9 |- ( ph -> ( U .(+) R ) e. S )
36	24, 35	sseldd 3156	. . . . . . . 8 |- ( ph -> ( U .(+) R ) e. (SubGrp ` W ) )
37		lsatcvat3.l	. . . . . . . 8 |- ( ph -> Q C_ ( U .(+) R ) )
38	2	lsmless2 14934	. . . . . . . 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 1187	. . . . . . 7 |- ( ph -> ( ( U .(+) R ) .(+) Q ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
40	33, 39	eqsstrd 3187	. . . . . 6 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) )
41	2	lsmidm 14936	. . . . . . 7 |- ( ( U .(+) R ) e. (SubGrp ` W ) -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) )
42	36, 41	syl 17	. . . . . 6 |- ( ph -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) )
43	40, 42	sseqtrd 3189	. . . . 5 |- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( U .(+) R ) )
44	24, 14	sseldd 3156	. . . . . 6 |- ( ph -> ( Q .(+) R ) e. (SubGrp ` W ) )
45	2	lsmub2 14931	. . . . . . 7 |- ( ( Q e. (SubGrp ` W ) /\ R e. (SubGrp ` W ) ) -> R C_ ( Q .(+) R ) )
46	25, 26, 45	syl2anc 645	. . . . . 6 |- ( ph -> R C_ ( Q .(+) R ) )
47	2	lsmless2 14934	. . . . . 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 1187	. . . . 5 |- ( ph -> ( U .(+) R ) C_ ( U .(+) ( Q .(+) R ) ) )
49	43, 48	eqssd 3171	. . . 4 |- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) R ) )
50	20, 49	breqtrrd 4023	. . 3 |- ( ph -> U ( <oLL ` W ) ( U .(+) ( Q .(+) R ) ) )
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 28477	. 2 |- ( ph -> ( U i^i ( Q .(+) R ) ) ( <oLL ` W ) ( Q .(+) R ) )
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 28491	1 |- ( ph -> ( U i^i ( Q .(+) R ) ) e. A )

Syntax hints:   -. wn 5   -> wi 6   = wceq 1619   e. wcel 1621   =/= wne 2421   i^i cin 3126   C_ wss 3127   class class class wbr 3997   ` cfv 4673  (class class class)co 5792  SubGrpcsubg 14578   LSSumclsm 14908   Abelcabel 15053   LModclmod 15590   LSubSpclss 15652   LVecclvec 15818  LSAtomsclsa 28414   <oL_L clcv 28458

This theorem is referenced by:  l1cvat  28495

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-0g 13367  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-grp 14452  df-minusg 14453  df-sbg 14454  df-subg 14581  df-cntz 14756  df-oppg 14782  df-lsm 14910  df-cmn 15054  df-abl 15055  df-mgp 15289  df-ring 15303  df-ur 15305  df-oppr 15368  df-dvdsr 15386  df-unit 15387  df-invr 15417  df-drng 15477  df-lmod 15592  df-lss 15653  df-lsp 15692  df-lvec 15819  df-lsatoms 28416  df-lcv 28459