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Mirrors > Home > MPE Home > Th. List > lvecindp | Structured version Visualization version GIF version |
Description: Compute the 𝑋 coefficient in a sum with an independent vector 𝑋 (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions 𝑌 and 𝑍 (second conjunct). Typically, 𝑈 is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 19-Jul-2022.) |
Ref | Expression |
---|---|
lvecindp.v | ⊢ 𝑉 = (Base‘𝑊) |
lvecindp.p | ⊢ + = (+g‘𝑊) |
lvecindp.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lvecindp.k | ⊢ 𝐾 = (Base‘𝐹) |
lvecindp.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lvecindp.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lvecindp.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lvecindp.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lvecindp.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lvecindp.n | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) |
lvecindp.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
lvecindp.z | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
lvecindp.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lvecindp.b | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
lvecindp.e | ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝑌) = ((𝐵 · 𝑋) + 𝑍)) |
Ref | Expression |
---|---|
lvecindp | ⊢ (𝜑 → (𝐴 = 𝐵 ∧ 𝑌 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecindp.p | . . . 4 ⊢ + = (+g‘𝑊) | |
2 | eqid 2798 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
3 | eqid 2798 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
4 | lvecindp.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 19871 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
7 | lvecindp.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
8 | lvecindp.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
9 | eqid 2798 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
10 | 8, 9 | lspsnsubg 19745 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑊)‘{𝑋}) ∈ (SubGrp‘𝑊)) |
11 | 6, 7, 10 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((LSpan‘𝑊)‘{𝑋}) ∈ (SubGrp‘𝑊)) |
12 | lvecindp.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊) | |
13 | 12 | lsssssubg 19723 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
14 | 6, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
15 | lvecindp.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
16 | 14, 15 | sseldd 3916 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
17 | lvecindp.n | . . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
18 | 8, 2, 9, 12, 4, 15, 7, 17 | lspdisj 19890 | . . . 4 ⊢ (𝜑 → (((LSpan‘𝑊)‘{𝑋}) ∩ 𝑈) = {(0g‘𝑊)}) |
19 | lmodabl 19674 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
20 | 6, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
21 | 3, 20, 11, 16 | ablcntzd 18970 | . . . 4 ⊢ (𝜑 → ((LSpan‘𝑊)‘{𝑋}) ⊆ ((Cntz‘𝑊)‘𝑈)) |
22 | lvecindp.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
23 | lvecindp.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
24 | lvecindp.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
25 | lvecindp.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
26 | 8, 22, 23, 24, 9, 6, 25, 7 | lspsneli 19766 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ ((LSpan‘𝑊)‘{𝑋})) |
27 | lvecindp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
28 | 8, 22, 23, 24, 9, 6, 27, 7 | lspsneli 19766 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝑋) ∈ ((LSpan‘𝑊)‘{𝑋})) |
29 | lvecindp.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
30 | lvecindp.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
31 | lvecindp.e | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝑌) = ((𝐵 · 𝑋) + 𝑍)) | |
32 | 1, 2, 3, 11, 16, 18, 21, 26, 28, 29, 30, 31 | subgdisj1 18809 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑋)) |
33 | 2, 12, 6, 15, 17 | lssvneln0 19716 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ (0g‘𝑊)) |
34 | 8, 22, 23, 24, 2, 4, 25, 27, 7, 33 | lvecvscan2 19877 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
35 | 32, 34 | mpbid 235 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
36 | 1, 2, 3, 11, 16, 18, 21, 26, 28, 29, 30, 31 | subgdisj2 18810 | . 2 ⊢ (𝜑 → 𝑌 = 𝑍) |
37 | 35, 36 | jca 515 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ 𝑌 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 SubGrpcsubg 18265 Cntzccntz 18437 Abelcabl 18899 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LVecclvec 19867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 |
This theorem is referenced by: baerlem3lem1 39003 baerlem5alem1 39004 baerlem5blem1 39005 |
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