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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 32424 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 2734 | . 2 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
5 | lsatcvat3.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 21122 | . . . 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 38978 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
11 | lsatcvat3.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 1, 3, 7, 11 | lsatlssel 38978 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
13 | 1, 2 | lsmcl 21099 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
14 | 7, 10, 12, 13 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
15 | 1 | lssincl 20980 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆) |
16 | 7, 8, 14, 15 | syl3anc 1370 | . 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 39022 | . . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅))) |
20 | 18, 19 | mpbid 232 | . . . 4 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅)) |
21 | lmodabl 20923 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
22 | 7, 21 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel) |
23 | 1 | lsssssubg 20973 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
24 | 7, 23 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
25 | 24, 10 | sseldd 3995 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
26 | 24, 12 | sseldd 3995 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
27 | 2 | lsmcom 19890 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
28 | 22, 25, 26, 27 | syl3anc 1370 | . . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
29 | 28 | oveq2d 7446 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
30 | 24, 8 | sseldd 3995 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
31 | 2 | lsmass 19701 | . . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
32 | 30, 26, 25, 31 | syl3anc 1370 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
33 | 29, 32 | eqtr4d 2777 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄)) |
34 | 1, 2 | lsmcl 21099 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
35 | 7, 8, 12, 34 | syl3anc 1370 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
36 | 24, 35 | sseldd 3995 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
37 | lsatcvat3.l | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) | |
38 | 2 | lsmless2 19693 | . . . . . . . 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 1370 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
40 | 33, 39 | eqsstrd 4033 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
41 | 2 | lsmidm 19695 | . . . . . . 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 4035 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅)) |
44 | 24, 14 | sseldd 3995 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
45 | 2 | lsmub2 19690 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
46 | 25, 26, 45 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
47 | 2 | lsmless2 19693 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
48 | 30, 44, 46, 47 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
49 | 43, 48 | eqssd 4012 | . . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅)) |
50 | 20, 49 | breqtrrd 5175 | . . 3 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
51 | 1, 2, 4, 7, 8, 14, 50 | lcvexchlem4 39018 | . 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖L ‘𝑊)(𝑄 ⊕ 𝑅)) |
52 | 1, 2, 3, 4, 5, 16, 9, 11, 17, 51 | lsatcvat2 39032 | 1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∩ cin 3961 ⊆ wss 3962 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 SubGrpcsubg 19150 LSSumclsm 19666 Abelcabl 19813 LModclmod 20874 LSubSpclss 20946 LVecclvec 21118 LSAtomsclsa 38955 ⋖L clcv 38999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-oppg 19376 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lcv 39000 |
This theorem is referenced by: l1cvat 39036 |
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