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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 32415 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 2737 | . 2 ⊢ ( ⋖L ‘𝑊) = ( ⋖L ‘𝑊) | |
| 5 | lsatcvat3.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
| 6 | lveclmod 21105 | . . . 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 38998 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
| 11 | lsatcvat3.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
| 12 | 1, 3, 7, 11 | lsatlssel 38998 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
| 13 | 1, 2 | lsmcl 21082 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
| 14 | 7, 10, 12, 13 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
| 15 | 1 | lssincl 20963 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑄 ⊕ 𝑅) ∈ 𝑆) → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝑆) |
| 16 | 7, 8, 14, 15 | syl3anc 1373 | . 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 39042 | . . . . 5 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑈 ↔ 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅))) |
| 20 | 18, 19 | mpbid 232 | . . . 4 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ 𝑅)) |
| 21 | lmodabl 20907 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
| 22 | 7, 21 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel) |
| 23 | 1 | lsssssubg 20956 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
| 24 | 7, 23 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
| 25 | 24, 10 | sseldd 3984 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
| 26 | 24, 12 | sseldd 3984 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
| 27 | 2 | lsmcom 19876 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
| 28 | 22, 25, 26, 27 | syl3anc 1373 | . . . . . . . . 9 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
| 29 | 28 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
| 30 | 24, 8 | sseldd 3984 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
| 31 | 2 | lsmass 19687 | . . . . . . . . 9 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑄 ∈ (SubGrp‘𝑊)) → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
| 32 | 30, 26, 25, 31 | syl3anc 1373 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) = (𝑈 ⊕ (𝑅 ⊕ 𝑄))) |
| 33 | 29, 32 | eqtr4d 2780 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = ((𝑈 ⊕ 𝑅) ⊕ 𝑄)) |
| 34 | 1, 2 | lsmcl 21082 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
| 35 | 7, 8, 12, 34 | syl3anc 1373 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ 𝑆) |
| 36 | 24, 35 | sseldd 3984 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
| 37 | lsatcvat3.l | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅)) | |
| 38 | 2 | lsmless2 19679 | . . . . . . . 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 1373 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ⊕ 𝑅) ⊕ 𝑄) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
| 40 | 33, 39 | eqsstrd 4018 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ ((𝑈 ⊕ 𝑅) ⊕ (𝑈 ⊕ 𝑅))) |
| 41 | 2 | lsmidm 19681 | . . . . . . 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 4020 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) ⊆ (𝑈 ⊕ 𝑅)) |
| 44 | 24, 14 | sseldd 3984 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊)) |
| 45 | 2 | lsmub2 19676 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
| 46 | 25, 26, 45 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑅 ⊆ (𝑄 ⊕ 𝑅)) |
| 47 | 2 | lsmless2 19679 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑄 ⊕ 𝑅) ∈ (SubGrp‘𝑊) ∧ 𝑅 ⊆ (𝑄 ⊕ 𝑅)) → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
| 48 | 30, 44, 46, 47 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑅) ⊆ (𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
| 49 | 43, 48 | eqssd 4001 | . . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑄 ⊕ 𝑅)) = (𝑈 ⊕ 𝑅)) |
| 50 | 20, 49 | breqtrrd 5171 | . . 3 ⊢ (𝜑 → 𝑈( ⋖L ‘𝑊)(𝑈 ⊕ (𝑄 ⊕ 𝑅))) |
| 51 | 1, 2, 4, 7, 8, 14, 50 | lcvexchlem4 39038 | . 2 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅))( ⋖L ‘𝑊)(𝑄 ⊕ 𝑅)) |
| 52 | 1, 2, 3, 4, 5, 16, 9, 11, 17, 51 | lsatcvat2 39052 | 1 ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 SubGrpcsubg 19138 LSSumclsm 19652 Abelcabl 19799 LModclmod 20858 LSubSpclss 20929 LVecclvec 21101 LSAtomsclsa 38975 ⋖L clcv 39019 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lcv 39020 |
| This theorem is referenced by: l1cvat 39056 |
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