lsatcvat3 - Mathbox for Norm Megill

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Theorem lsatcvat3 40375

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 33055 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 2763	. 2 ⊢ ( ⋖_L ‘𝑊) = ( ⋖_L ‘𝑊)
5		lsatcvat3.w	. 2 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 21432	. . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
7	5, 6	syl 18	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
8		lsatcvat3.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
9		lsatcvat3.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10	1, 3, 7, 9	lsatlssel 40320	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
11		lsatcvat3.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	1, 3, 7, 11	lsatlssel 40320	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
13	1, 2	lsmcl 21409	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆)
14	7, 10, 12, 13	syl3anc 1399	. . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆)
15	1	lssincl 21291	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆)
16	7, 8, 14, 15	syl3anc 1399	. 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 40364	. . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅)))
20	18, 19	mpbid 235	. . . 4 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅))
21		lmodabl 21235	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
22	7, 21	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
23	1	lsssssubg 21284	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
24	7, 23	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
25	24, 10	sseldd 3914	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
26	24, 12	sseldd 3914	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
27	2	lsmcom 20123	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
28	22, 25, 26, 27	syl3anc 1399	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
29	28	oveq2d 7467	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
30	24, 8	sseldd 3914	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
31	2	lsmass 19934	. . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
32	30, 26, 25, 31	syl3anc 1399	. . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
33	29, 32	eqtr4d 2801	. . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄))
34	1, 2	lsmcl 21409	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆)
35	7, 8, 12, 34	syl3anc 1399	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆)
36	24, 35	sseldd 3914	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
37		lsatcvat3.l	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))
38	2	lsmless2 19926	. . . . . . . 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 1399	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
40	33, 39	eqsstrd 3947	. . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
41	2	lsmidm 19928	. . . . . . 7 ⊢ ((𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊) → ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
42	36, 41	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
43	40, 42	sseqtrd 3949	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅))
44	24, 14	sseldd 3914	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
45	2	lsmub2 19923	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
46	25, 26, 45	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
47	2	lsmless2 19926	. . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
48	30, 44, 46, 47	syl3anc 1399	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅)))
49	43, 48	eqssd 3930	. . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
50	20, 49	breqtrrd 5139	. . 3 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅)))
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 40360	. 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖_L ‘𝑊)(𝑄 ⊕ 𝑅))
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 40374	1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ⊆ wss 3900 ∩ cin 4092 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 SubGrpcsubg 19380 LSSumclsm 19899 Abelcabl 20046 LModclmod 21186 LSubSpclss 21257 LVecclvec 21428 LSAtomsclsa 40297 ⋖_L clcv 40341

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-tpos 8281 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-0g 17662 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-subg 19383 df-cntz 19582 df-oppg 19611 df-lsm 19901 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-drng 21033 df-lmod 21188 df-lss 21258 df-lsp 21298 df-lvec 21429 df-lsatoms 40299 df-lcv 40342

This theorem is used by: l1cvat 40378

W3C validator