Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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.) |
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 2610 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
3 | eqid 2610 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
4 | lvecindp.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 18927 | . . . . . 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 2610 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
10 | 8, 9 | lspsnsubg 18801 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑊)‘{𝑋}) ∈ (SubGrp‘𝑊)) |
11 | 6, 7, 10 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ((LSpan‘𝑊)‘{𝑋}) ∈ (SubGrp‘𝑊)) |
12 | lvecindp.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊) | |
13 | 12 | lsssssubg 18779 | . . . . . 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 3569 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
17 | lvecindp.n | . . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
18 | 8, 2, 9, 12, 4, 15, 7, 17 | lspdisj 18946 | . . . 4 ⊢ (𝜑 → (((LSpan‘𝑊)‘{𝑋}) ∩ 𝑈) = {(0g‘𝑊)}) |
19 | lmodabl 18733 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
20 | 6, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
21 | 3, 20, 11, 16 | ablcntzd 18083 | . . . 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 18822 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ ((LSpan‘𝑊)‘{𝑋})) |
27 | lvecindp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
28 | 8, 22, 23, 24, 9, 6, 27, 7 | lspsneli 18822 | . . . 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 17927 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑋)) |
33 | 8, 2, 12, 6, 15, 7, 17 | lssneln0 18773 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ {(0g‘𝑊)})) |
34 | eldifsni 4261 | . . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ {(0g‘𝑊)}) → 𝑋 ≠ (0g‘𝑊)) | |
35 | 33, 34 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ (0g‘𝑊)) |
36 | 8, 22, 23, 24, 2, 4, 25, 27, 7, 35 | lvecvscan2 18933 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
37 | 32, 36 | mpbid 221 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
38 | 1, 2, 3, 11, 16, 18, 21, 26, 28, 29, 30, 31 | subgdisj2 17928 | . 2 ⊢ (𝜑 → 𝑌 = 𝑍) |
39 | 37, 38 | jca 553 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ 𝑌 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 SubGrpcsubg 17411 Cntzccntz 17571 Abelcabl 18017 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 |
This theorem is referenced by: baerlem3lem1 36014 baerlem5alem1 36015 baerlem5blem1 36016 |
Copyright terms: Public domain | W3C validator |