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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat3 | Structured version Visualization version GIF version |
Description: A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30100 analog.) (Contributed by NM, 11-Jan-2015.) |
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)) |
Ref | Expression |
---|---|
lsatcvat3 | ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) |
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 2818 | . 2 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
5 | lsatcvat3.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 19807 | . . . 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 36013 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
11 | lsatcvat3.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 1, 3, 7, 11 | lsatlssel 36013 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
13 | 1, 2 | lsmcl 19784 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
14 | 7, 10, 12, 13 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
15 | 1 | lssincl 19666 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆) |
16 | 7, 8, 14, 15 | syl3anc 1363 | . 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 36057 | . . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅))) |
20 | 18, 19 | mpbid 233 | . . . 4 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅)) |
21 | lmodabl 19610 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
22 | 7, 21 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel) |
23 | 1 | lsssssubg 19659 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
24 | 7, 23 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
25 | 24, 10 | sseldd 3965 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
26 | 24, 12 | sseldd 3965 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
27 | 2 | lsmcom 18907 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
28 | 22, 25, 26, 27 | syl3anc 1363 | . . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
29 | 28 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
30 | 24, 8 | sseldd 3965 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
31 | 2 | lsmass 18724 | . . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
32 | 30, 26, 25, 31 | syl3anc 1363 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
33 | 29, 32 | eqtr4d 2856 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄)) |
34 | 1, 2 | lsmcl 19784 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
35 | 7, 8, 12, 34 | syl3anc 1363 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
36 | 24, 35 | sseldd 3965 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
37 | lsatcvat3.l | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) | |
38 | 2 | lsmless2 18715 | . . . . . . . 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 1363 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
40 | 33, 39 | eqsstrd 4002 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
41 | 2 | lsmidm 18717 | . . . . . . 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 4004 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅)) |
44 | 24, 14 | sseldd 3965 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
45 | 2 | lsmub2 18712 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
46 | 25, 26, 45 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
47 | 2 | lsmless2 18715 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
48 | 30, 44, 46, 47 | syl3anc 1363 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
49 | 43, 48 | eqssd 3981 | . . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅)) |
50 | 20, 49 | breqtrrd 5085 | . . 3 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
51 | 1, 2, 4, 7, 8, 14, 50 | lcvexchlem4 36053 | . 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖L ‘𝑊)(𝑄 ⊕ 𝑅)) |
52 | 1, 2, 3, 4, 5, 16, 9, 11, 17, 51 | lsatcvat2 36067 | 1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 SubGrpcsubg 18211 LSSumclsm 18688 Abelcabl 18836 LModclmod 19563 LSubSpclss 19632 LVecclvec 19803 LSAtomsclsa 35990 ⋖L clcv 36034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lcv 36035 |
This theorem is referenced by: l1cvat 36071 |
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