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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmsatcv | Structured version Visualization version GIF version |
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 29116 analog.) Explicit atom version of lsmcv 19607. (Contributed by NM, 29-Oct-2014.) |
Ref | Expression |
---|---|
lsmsatcv.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsmsatcv.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsmsatcv.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsmsatcv.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsmsatcv.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lsmsatcv.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsmsatcv.x | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
lsmsatcv | ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmsatcv.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lsmsatcv.x | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
3 | eqid 2797 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2797 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | lsmsatcv.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
6 | 3, 4, 5 | islsati 35682 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑄 ∈ 𝐴) → ∃𝑣 ∈ (Base‘𝑊)𝑄 = ((LSpan‘𝑊)‘{𝑣})) |
7 | 1, 2, 6 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∃𝑣 ∈ (Base‘𝑊)𝑄 = ((LSpan‘𝑊)‘{𝑣})) |
8 | lsmsatcv.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
9 | lsmsatcv.p | . . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 1 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑊 ∈ LVec) |
11 | lsmsatcv.t | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
12 | 11 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑇 ∈ 𝑆) |
13 | lsmsatcv.u | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
14 | 13 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑈 ∈ 𝑆) |
15 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ (Base‘𝑊)) | |
16 | 3, 8, 4, 9, 10, 12, 14, 15 | lsmcv 19607 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) |
17 | 16 | 3expib 1115 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
18 | 17 | 3adant3 1125 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
19 | oveq2 7031 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑇 ⊕ 𝑄) = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) | |
20 | 19 | sseq2d 3926 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 ⊆ (𝑇 ⊕ 𝑄) ↔ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
21 | 20 | anbi2d 628 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) ↔ (𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))))) |
22 | 19 | eqeq2d 2807 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 = (𝑇 ⊕ 𝑄) ↔ 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
23 | 21, 22 | imbi12d 346 | . . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) ↔ ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))))) |
24 | 23 | 3ad2ant3 1128 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) ↔ ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))))) |
25 | 18, 24 | mpbird 258 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄))) |
26 | 25 | rexlimdv3a 3251 | . . 3 ⊢ (𝜑 → (∃𝑣 ∈ (Base‘𝑊)𝑄 = ((LSpan‘𝑊)‘{𝑣}) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)))) |
27 | 7, 26 | mpd 15 | . 2 ⊢ (𝜑 → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄))) |
28 | 27 | 3impib 1109 | 1 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∃wrex 3108 ⊆ wss 3865 ⊊ wpss 3866 {csn 4478 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 LSSumclsm 18493 LSubSpclss 19397 LSpanclspn 19437 LVecclvec 19568 LSAtomsclsa 35662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-subg 18034 df-lsm 18495 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-drng 19198 df-lmod 19330 df-lss 19398 df-lsp 19438 df-lvec 19569 df-lsatoms 35664 |
This theorem is referenced by: dochsat 38071 |
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