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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 2821 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
4 | lvecindp.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 19878 | . . . . . 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 2821 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
10 | 8, 9 | lspsnsubg 19752 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑊)‘{𝑋}) ∈ (SubGrp‘𝑊)) |
11 | 6, 7, 10 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((LSpan‘𝑊)‘{𝑋}) ∈ (SubGrp‘𝑊)) |
12 | lvecindp.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊) | |
13 | 12 | lsssssubg 19730 | . . . . . 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 3968 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
17 | lvecindp.n | . . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
18 | 8, 2, 9, 12, 4, 15, 7, 17 | lspdisj 19897 | . . . 4 ⊢ (𝜑 → (((LSpan‘𝑊)‘{𝑋}) ∩ 𝑈) = {(0g‘𝑊)}) |
19 | lmodabl 19681 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
20 | 6, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
21 | 3, 20, 11, 16 | ablcntzd 18977 | . . . 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 19773 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ ((LSpan‘𝑊)‘{𝑋})) |
27 | lvecindp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
28 | 8, 22, 23, 24, 9, 6, 27, 7 | lspsneli 19773 | . . . 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 18817 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑋)) |
33 | 2, 12, 6, 15, 17 | lssvneln0 19723 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ (0g‘𝑊)) |
34 | 8, 22, 23, 24, 2, 4, 25, 27, 7, 33 | lvecvscan2 19884 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) |
35 | 32, 34 | mpbid 234 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
36 | 1, 2, 3, 11, 16, 18, 21, 26, 28, 29, 30, 31 | subgdisj2 18818 | . 2 ⊢ (𝜑 → 𝑌 = 𝑍) |
37 | 35, 36 | jca 514 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ 𝑌 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 {csn 4567 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 SubGrpcsubg 18273 Cntzccntz 18445 Abelcabl 18907 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LVecclvec 19874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 |
This theorem is referenced by: baerlem3lem1 38858 baerlem5alem1 38859 baerlem5blem1 38860 |
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