lsatcvat3 - Mathbox for Norm Megill

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Theorem lsatcvat3 39161

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32376 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	⊢ (𝜑 → 𝑊 ∈ LVec)
lsatcvat3.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsatcvat3.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsatcvat3.r	⊢ (𝜑 → 𝑅 ∈ 𝐴)
lsatcvat3.n	⊢ (𝜑 → 𝑄 ≠ 𝑅)
lsatcvat3.m	⊢ (𝜑 → ¬ 𝑅 ⊆ 𝑈)
lsatcvat3.l	⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))

**Assertion**
Ref	Expression
lsatcvat3	⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 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 2731	. 2 ⊢ ( ⋖_L ‘𝑊) = ( ⋖_L ‘𝑊)
5		lsatcvat3.w	. 2 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 21040	. . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		lsatcvat3.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
9		lsatcvat3.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10	1, 3, 7, 9	lsatlssel 39106	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
11		lsatcvat3.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	1, 3, 7, 11	lsatlssel 39106	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
13	1, 2	lsmcl 21017	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆)
14	7, 10, 12, 13	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆)
15	1	lssincl 20898	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆)
16	7, 8, 14, 15	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆)
17		lsatcvat3.n	. 2 ⊢ (𝜑 → 𝑄 ≠ 𝑅)
18		lsatcvat3.m	. . . . 5 ⊢ (𝜑 → ¬ 𝑅 ⊆ 𝑈)
19	1, 2, 3, 4, 5, 8, 11	lcv1 39150	. . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅)))
20	18, 19	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅))
21		lmodabl 20842	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
22	7, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
23	1	lsssssubg 20891	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
24	7, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
25	24, 10	sseldd 3930	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
26	24, 12	sseldd 3930	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
27	2	lsmcom 19770	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
28	22, 25, 26, 27	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
29	28	oveq2d 7362	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
30	24, 8	sseldd 3930	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
31	2	lsmass 19581	. . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
32	30, 26, 25, 31	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
33	29, 32	eqtr4d 2769	. . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄))
34	1, 2	lsmcl 21017	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆)
35	7, 8, 12, 34	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆)
36	24, 35	sseldd 3930	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
37		lsatcvat3.l	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))
38	2	lsmless2 19573	. . . . . . . 8 ⊢ (((𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑄 ⊆ (𝑈 ⊕ 𝑅)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
39	36, 36, 37, 38	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
40	33, 39	eqsstrd 3964	. . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
41	2	lsmidm 19575	. . . . . . 7 ⊢ ((𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊) → ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
42	36, 41	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
43	40, 42	sseqtrd 3966	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅))
44	24, 14	sseldd 3930	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
45	2	lsmub2 19570	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
46	25, 26, 45	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
47	2	lsmless2 19573	. . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
48	30, 44, 46, 47	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
49	43, 48	eqssd 3947	. . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
50	20, 49	breqtrrd 5117	. . 3 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅)))
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 39146	. 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖_L ‘𝑊)(𝑄 ⊕ 𝑅))
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 39160	1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∩ cin 3896 ⊆ wss 3897 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 SubGrpcsubg 19033 LSSumclsm 19546 Abelcabl 19693 LModclmod 20793 LSubSpclss 20864 LVecclvec 21036 LSAtomsclsa 39083 ⋖_L clcv 39127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19229 df-oppg 19258 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-drng 20646 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lvec 21037 df-lsatoms 39085 df-lcv 39128

This theorem is referenced by: l1cvat 39164

W3C validator