lsatcvat3 - Mathbox for Norm Megill

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Theorem lsatcvat3 39515

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32485 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 2737	. 2 ⊢ ( ⋖_L ‘𝑊) = ( ⋖_L ‘𝑊)
5		lsatcvat3.w	. 2 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 21096	. . . 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 39460	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
11		lsatcvat3.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	1, 3, 7, 11	lsatlssel 39460	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
13	1, 2	lsmcl 21073	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆)
14	7, 10, 12, 13	syl3anc 1374	. . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆)
15	1	lssincl 20954	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆)
16	7, 8, 14, 15	syl3anc 1374	. 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 39504	. . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅)))
20	18, 19	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅))
21		lmodabl 20898	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
22	7, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
23	1	lsssssubg 20947	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
24	7, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
25	24, 10	sseldd 3923	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
26	24, 12	sseldd 3923	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
27	2	lsmcom 19827	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
28	22, 25, 26, 27	syl3anc 1374	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
29	28	oveq2d 7377	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
30	24, 8	sseldd 3923	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
31	2	lsmass 19638	. . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
32	30, 26, 25, 31	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
33	29, 32	eqtr4d 2775	. . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄))
34	1, 2	lsmcl 21073	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆)
35	7, 8, 12, 34	syl3anc 1374	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆)
36	24, 35	sseldd 3923	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
37		lsatcvat3.l	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))
38	2	lsmless2 19630	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
40	33, 39	eqsstrd 3957	. . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
41	2	lsmidm 19632	. . . . . . 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 3959	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅))
44	24, 14	sseldd 3923	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
45	2	lsmub2 19627	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
46	25, 26, 45	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
47	2	lsmless2 19630	. . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
48	30, 44, 46, 47	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
49	43, 48	eqssd 3940	. . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
50	20, 49	breqtrrd 5114	. . 3 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅)))
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 39500	. 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖_L ‘𝑊)(𝑄 ⊕ 𝑅))
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 39514	1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 ‘cfv 6493 (class class class)co 7361 SubGrpcsubg 19090 LSSumclsm 19603 Abelcabl 19750 LModclmod 20849 LSubSpclss 20920 LVecclvec 21092 LSAtomsclsa 39437 ⋖_L clcv 39481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-0g 17398 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-cntz 19286 df-oppg 19315 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-drng 20702 df-lmod 20851 df-lss 20921 df-lsp 20961 df-lvec 21093 df-lsatoms 39439 df-lcv 39482

This theorem is referenced by: l1cvat 39518

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