lsatcvat3 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat3		Structured version Visualization version GIF version

Theorem lsatcvat3 36182

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30167 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 2821	. 2 ⊢ ( ⋖_L ‘𝑊) = ( ⋖_L ‘𝑊)
5		lsatcvat3.w	. 2 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 19872	. . . 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 36127	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
11		lsatcvat3.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	1, 3, 7, 11	lsatlssel 36127	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
13	1, 2	lsmcl 19849	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆)
14	7, 10, 12, 13	syl3anc 1367	. . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆)
15	1	lssincl 19731	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆)
16	7, 8, 14, 15	syl3anc 1367	. 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 36171	. . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅)))
20	18, 19	mpbid 234	. . . 4 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅))
21		lmodabl 19675	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
22	7, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
23	1	lsssssubg 19724	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
24	7, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
25	24, 10	sseldd 3968	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
26	24, 12	sseldd 3968	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
27	2	lsmcom 18972	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
28	22, 25, 26, 27	syl3anc 1367	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
29	28	oveq2d 7166	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
30	24, 8	sseldd 3968	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
31	2	lsmass 18789	. . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
32	30, 26, 25, 31	syl3anc 1367	. . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
33	29, 32	eqtr4d 2859	. . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄))
34	1, 2	lsmcl 19849	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆)
35	7, 8, 12, 34	syl3anc 1367	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆)
36	24, 35	sseldd 3968	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
37		lsatcvat3.l	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))
38	2	lsmless2 18780	. . . . . . . 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 1367	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
40	33, 39	eqsstrd 4005	. . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
41	2	lsmidm 18782	. . . . . . 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 4007	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅))
44	24, 14	sseldd 3968	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
45	2	lsmub2 18777	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
46	25, 26, 45	syl2anc 586	. . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
47	2	lsmless2 18780	. . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
48	30, 44, 46, 47	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
49	43, 48	eqssd 3984	. . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
50	20, 49	breqtrrd 5087	. . 3 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅)))
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 36167	. 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖_L ‘𝑊)(𝑄 ⊕ 𝑅))
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 36181	1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 SubGrpcsubg 18267 LSSumclsm 18753 Abelcabl 18901 LModclmod 19628 LSubSpclss 19697 LVecclvec 19868 LSAtomsclsa 36104 ⋖_L clcv 36148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lcv 36149

This theorem is referenced by: l1cvat 36185

W3C validator