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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 2738 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
3 | eqid 2738 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
4 | lvecindp.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 19997 | . . . . . 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 2738 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
10 | 8, 9 | lspsnsubg 19871 | . . . . 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 19849 | . . . . . 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 3878 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
17 | lvecindp.n | . . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
18 | 8, 2, 9, 12, 4, 15, 7, 17 | lspdisj 20016 | . . . 4 ⊢ (𝜑 → (((LSpan‘𝑊)‘{𝑋}) ∩ 𝑈) = {(0g‘𝑊)}) |
19 | lmodabl 19800 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
20 | 6, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
21 | 3, 20, 11, 16 | ablcntzd 19096 | . . . 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 19892 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ ((LSpan‘𝑊)‘{𝑋})) |
27 | lvecindp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
28 | 8, 22, 23, 24, 9, 6, 27, 7 | lspsneli 19892 | . . . 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 18935 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑋)) |
33 | 2, 12, 6, 15, 17 | lssvneln0 19842 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ (0g‘𝑊)) |
34 | 8, 22, 23, 24, 2, 4, 25, 27, 7, 33 | lvecvscan2 20003 | . . 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 18936 | . 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 1542 ∈ wcel 2114 ⊆ wss 3843 {csn 4516 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 +gcplusg 16668 Scalarcsca 16671 ·𝑠 cvsca 16672 0gc0g 16816 SubGrpcsubg 18391 Cntzccntz 18563 Abelcabl 19025 LModclmod 19753 LSubSpclss 19822 LSpanclspn 19862 LVecclvec 19993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-cntz 18565 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-drng 19623 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lvec 19994 |
This theorem is referenced by: baerlem3lem1 39344 baerlem5alem1 39345 baerlem5blem1 39346 |
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