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Mirrors > Home > MPE Home > Th. List > Mathboxes > l1cvpat | Structured version Visualization version GIF version |
Description: A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 36615 analog.) (Contributed by NM, 11-Jan-2015.) |
Ref | Expression |
---|---|
l1cvpat.v | ⊢ 𝑉 = (Base‘𝑊) |
l1cvpat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
l1cvpat.p | ⊢ ⊕ = (LSSum‘𝑊) |
l1cvpat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
l1cvpat.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
l1cvpat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
l1cvpat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
l1cvpat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
l1cvpat.l | ⊢ (𝜑 → 𝑈𝐶𝑉) |
l1cvpat.m | ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) |
Ref | Expression |
---|---|
l1cvpat | ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | l1cvpat.q | . . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
2 | l1cvpat.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | l1cvpat.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2824 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | eqid 2824 | . . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
6 | l1cvpat.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | 3, 4, 5, 6 | islsat 36131 | . . . 4 ⊢ (𝑊 ∈ LVec → (𝑄 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑣}))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑣}))) |
9 | 1, 8 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑣})) |
10 | l1cvpat.m | . 2 ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) | |
11 | eldifi 4106 | . . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)}) → 𝑣 ∈ 𝑉) | |
12 | l1cvpat.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
13 | lveclmod 19881 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
14 | 2, 13 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod) |
15 | 14 | 3ad2ant1 1129 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → 𝑊 ∈ LMod) |
16 | l1cvpat.u | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
17 | 16 | 3ad2ant1 1129 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → 𝑈 ∈ 𝑆) |
18 | simp2 1133 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → 𝑣 ∈ 𝑉) | |
19 | 3, 12, 4, 15, 17, 18 | lspsnel5 19770 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (𝑣 ∈ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑣}) ⊆ 𝑈)) |
20 | 19 | notbid 320 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (¬ 𝑣 ∈ 𝑈 ↔ ¬ ((LSpan‘𝑊)‘{𝑣}) ⊆ 𝑈)) |
21 | l1cvpat.p | . . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝑊) | |
22 | eqid 2824 | . . . . . . . . 9 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊) | |
23 | 2 | 3ad2ant1 1129 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → 𝑊 ∈ LVec) |
24 | l1cvpat.l | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈𝐶𝑉) | |
25 | l1cvpat.c | . . . . . . . . . . . 12 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
26 | 3, 12, 22, 25, 2 | islshpcv 36193 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 ∈ (LSHyp‘𝑊) ↔ (𝑈 ∈ 𝑆 ∧ 𝑈𝐶𝑉))) |
27 | 16, 24, 26 | mpbir2and 711 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (LSHyp‘𝑊)) |
28 | 27 | 3ad2ant1 1129 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → 𝑈 ∈ (LSHyp‘𝑊)) |
29 | 3, 4, 21, 22, 23, 28, 18 | lshpnelb 36124 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (¬ 𝑣 ∈ 𝑈 ↔ (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑣})) = 𝑉)) |
30 | 29 | biimpd 231 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (¬ 𝑣 ∈ 𝑈 → (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑣})) = 𝑉)) |
31 | 20, 30 | sylbird 262 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (¬ ((LSpan‘𝑊)‘{𝑣}) ⊆ 𝑈 → (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑣})) = 𝑉)) |
32 | sseq1 3995 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑄 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑣}) ⊆ 𝑈)) | |
33 | 32 | notbid 320 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (¬ 𝑄 ⊆ 𝑈 ↔ ¬ ((LSpan‘𝑊)‘{𝑣}) ⊆ 𝑈)) |
34 | oveq2 7167 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 ⊕ 𝑄) = (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑣}))) | |
35 | 34 | eqeq1d 2826 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → ((𝑈 ⊕ 𝑄) = 𝑉 ↔ (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑣})) = 𝑉)) |
36 | 33, 35 | imbi12d 347 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → ((¬ 𝑄 ⊆ 𝑈 → (𝑈 ⊕ 𝑄) = 𝑉) ↔ (¬ ((LSpan‘𝑊)‘{𝑣}) ⊆ 𝑈 → (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑣})) = 𝑉))) |
37 | 36 | 3ad2ant3 1131 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → ((¬ 𝑄 ⊆ 𝑈 → (𝑈 ⊕ 𝑄) = 𝑉) ↔ (¬ ((LSpan‘𝑊)‘{𝑣}) ⊆ 𝑈 → (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑣})) = 𝑉))) |
38 | 31, 37 | mpbird 259 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉 ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (¬ 𝑄 ⊆ 𝑈 → (𝑈 ⊕ 𝑄) = 𝑉)) |
39 | 38 | 3exp 1115 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ 𝑉 → (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (¬ 𝑄 ⊆ 𝑈 → (𝑈 ⊕ 𝑄) = 𝑉)))) |
40 | 11, 39 | syl5 34 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)}) → (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (¬ 𝑄 ⊆ 𝑈 → (𝑈 ⊕ 𝑄) = 𝑉)))) |
41 | 40 | rexlimdv 3286 | . 2 ⊢ (𝜑 → (∃𝑣 ∈ (𝑉 ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (¬ 𝑄 ⊆ 𝑈 → (𝑈 ⊕ 𝑄) = 𝑉))) |
42 | 9, 10, 41 | mp2d 49 | 1 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ∖ cdif 3936 ⊆ wss 3939 {csn 4570 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 0gc0g 16716 LSSumclsm 18762 LModclmod 19637 LSubSpclss 19706 LSpanclspn 19746 LVecclvec 19877 LSAtomsclsa 36114 LSHypclsh 36115 ⋖L clcv 36158 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lsatoms 36116 df-lshyp 36117 df-lcv 36159 |
This theorem is referenced by: l1cvat 36195 |
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