Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 29255 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 2622 | . 2 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
5 | lsatcvat3.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 19106 | . . . 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 34284 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
11 | lsatcvat3.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 1, 3, 7, 11 | lsatlssel 34284 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
13 | 1, 2 | lsmcl 19083 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
14 | 7, 10, 12, 13 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
15 | 1 | lssincl 18965 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆) |
16 | 7, 8, 14, 15 | syl3anc 1326 | . 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 34328 | . . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅))) |
20 | 18, 19 | mpbid 222 | . . . 4 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅)) |
21 | lmodabl 18910 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
22 | 7, 21 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel) |
23 | 1 | lsssssubg 18958 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
24 | 7, 23 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
25 | 24, 10 | sseldd 3604 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
26 | 24, 12 | sseldd 3604 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
27 | 2 | lsmcom 18261 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
28 | 22, 25, 26, 27 | syl3anc 1326 | . . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
29 | 28 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
30 | 24, 8 | sseldd 3604 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
31 | 2 | lsmass 18083 | . . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
32 | 30, 26, 25, 31 | syl3anc 1326 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
33 | 29, 32 | eqtr4d 2659 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄)) |
34 | 1, 2 | lsmcl 19083 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
35 | 7, 8, 12, 34 | syl3anc 1326 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
36 | 24, 35 | sseldd 3604 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
37 | lsatcvat3.l | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) | |
38 | 2 | lsmless2 18075 | . . . . . . . 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 1326 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
40 | 33, 39 | eqsstrd 3639 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
41 | 2 | lsmidm 18077 | . . . . . . 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 3641 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅)) |
44 | 24, 14 | sseldd 3604 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
45 | 2 | lsmub2 18072 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
46 | 25, 26, 45 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
47 | 2 | lsmless2 18075 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
48 | 30, 44, 46, 47 | syl3anc 1326 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
49 | 43, 48 | eqssd 3620 | . . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅)) |
50 | 20, 49 | breqtrrd 4681 | . . 3 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
51 | 1, 2, 4, 7, 8, 14, 50 | lcvexchlem4 34324 | . 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖L ‘𝑊)(𝑄 ⊕ 𝑅)) |
52 | 1, 2, 3, 4, 5, 16, 9, 11, 17, 51 | lsatcvat2 34338 | 1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 SubGrpcsubg 17588 LSSumclsm 18049 Abelcabl 18194 LModclmod 18863 LSubSpclss 18932 LVecclvec 19102 LSAtomsclsa 34261 ⋖L clcv 34305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lcv 34306 |
This theorem is referenced by: l1cvat 34342 |
Copyright terms: Public domain | W3C validator |