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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 30737 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 2739 | . 2 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
5 | lsatcvat3.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 20349 | . . . 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 36990 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
11 | lsatcvat3.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 1, 3, 7, 11 | lsatlssel 36990 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
13 | 1, 2 | lsmcl 20326 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
14 | 7, 10, 12, 13 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
15 | 1 | lssincl 20208 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆) |
16 | 7, 8, 14, 15 | syl3anc 1369 | . 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 37034 | . . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅))) |
20 | 18, 19 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅)) |
21 | lmodabl 20151 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
22 | 7, 21 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel) |
23 | 1 | lsssssubg 20201 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
24 | 7, 23 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
25 | 24, 10 | sseldd 3926 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
26 | 24, 12 | sseldd 3926 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
27 | 2 | lsmcom 19440 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
28 | 22, 25, 26, 27 | syl3anc 1369 | . . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
29 | 28 | oveq2d 7284 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
30 | 24, 8 | sseldd 3926 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
31 | 2 | lsmass 19256 | . . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
32 | 30, 26, 25, 31 | syl3anc 1369 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
33 | 29, 32 | eqtr4d 2782 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄)) |
34 | 1, 2 | lsmcl 20326 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
35 | 7, 8, 12, 34 | syl3anc 1369 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
36 | 24, 35 | sseldd 3926 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
37 | lsatcvat3.l | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) | |
38 | 2 | lsmless2 19247 | . . . . . . . 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 1369 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
40 | 33, 39 | eqsstrd 3963 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
41 | 2 | lsmidm 19249 | . . . . . . 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 3965 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅)) |
44 | 24, 14 | sseldd 3926 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
45 | 2 | lsmub2 19244 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
46 | 25, 26, 45 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
47 | 2 | lsmless2 19247 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
48 | 30, 44, 46, 47 | syl3anc 1369 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
49 | 43, 48 | eqssd 3942 | . . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅)) |
50 | 20, 49 | breqtrrd 5106 | . . 3 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
51 | 1, 2, 4, 7, 8, 14, 50 | lcvexchlem4 37030 | . 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖L ‘𝑊)(𝑄 ⊕ 𝑅)) |
52 | 1, 2, 3, 4, 5, 16, 9, 11, 17, 51 | lsatcvat2 37044 | 1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ∩ cin 3890 ⊆ wss 3891 class class class wbr 5078 ‘cfv 6430 (class class class)co 7268 SubGrpcsubg 18730 LSSumclsm 19220 Abelcabl 19368 LModclmod 20104 LSubSpclss 20174 LVecclvec 20345 LSAtomsclsa 36967 ⋖L clcv 37011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-0g 17133 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-cntz 18904 df-oppg 18931 df-lsm 19222 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-drng 19974 df-lmod 20106 df-lss 20175 df-lsp 20215 df-lvec 20346 df-lsatoms 36969 df-lcv 37012 |
This theorem is referenced by: l1cvat 37048 |
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