lsatcvat3 - Mathbox for Norm Megill

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Theorem lsatcvat3 39498

Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32467 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 2736	. 2 ⊢ ( ⋖_L ‘𝑊) = ( ⋖_L ‘𝑊)
5		lsatcvat3.w	. 2 ⊢ (𝜑 → 𝑊 ∈ LVec)
6		lveclmod 21101	. . . 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 39443	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
11		lsatcvat3.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	1, 3, 7, 11	lsatlssel 39443	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
13	1, 2	lsmcl 21078	. . . 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 20960	. . 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 39487	. . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅)))
20	18, 19	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ 𝑅))
21		lmodabl 20904	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
22	7, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
23	1	lsssssubg 20953	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
24	7, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
25	24, 10	sseldd 3922	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
26	24, 12	sseldd 3922	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
27	2	lsmcom 19833	. . . . . . . . . 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 7383	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄)))
30	24, 8	sseldd 3922	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
31	2	lsmass 19644	. . . . . . . . 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 2774	. . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄))
34	1, 2	lsmcl 21078	. . . . . . . . . 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 3922	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
37		lsatcvat3.l	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))
38	2	lsmless2 19636	. . . . . . . 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 3956	. . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅)))
41	2	lsmidm 19638	. . . . . . 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 3958	. . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅))
44	24, 14	sseldd 3922	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊))
45	2	lsmub2 19633	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
46	25, 26, 45	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅))
47	2	lsmless2 19636	. . . . . 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 3939	. . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅))
50	20, 49	breqtrrd 5113	. . 3 ⊢ (𝜑 → 𝑈( ⋖_L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅)))
51	1, 2, 4, 7, 8, 14, 50	lcvexchlem4 39483	. 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖_L ‘𝑊)(𝑄 ⊕ 𝑅))
52	1, 2, 3, 4, 5, 16, 9, 11, 17, 51	lsatcvat2 39497	1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∩ cin 3888 ⊆ wss 3889 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 SubGrpcsubg 19096 LSSumclsm 19609 Abelcabl 19756 LModclmod 20855 LSubSpclss 20926 LVecclvec 21097 LSAtomsclsa 39420 ⋖_L clcv 39464

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-0g 17404 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-cntz 19292 df-oppg 19321 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lvec 21098 df-lsatoms 39422 df-lcv 39465

This theorem is referenced by: l1cvat 39501

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