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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 2823 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
3 | eqid 2823 | . . . 4 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
4 | lvecindp.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 19880 | . . . . . 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 2823 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
10 | 8, 9 | lspsnsubg 19754 | . . . . 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 19732 | . . . . . 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 3970 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
17 | lvecindp.n | . . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
18 | 8, 2, 9, 12, 4, 15, 7, 17 | lspdisj 19899 | . . . 4 ⊢ (𝜑 → (((LSpan‘𝑊)‘{𝑋}) ∩ 𝑈) = {(0g‘𝑊)}) |
19 | lmodabl 19683 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
20 | 6, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
21 | 3, 20, 11, 16 | ablcntzd 18979 | . . . 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 19775 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ ((LSpan‘𝑊)‘{𝑋})) |
27 | lvecindp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
28 | 8, 22, 23, 24, 9, 6, 27, 7 | lspsneli 19775 | . . . 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 18819 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑋)) |
33 | 2, 12, 6, 15, 17 | lssvneln0 19725 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ (0g‘𝑊)) |
34 | 8, 22, 23, 24, 2, 4, 25, 27, 7, 33 | lvecvscan2 19886 | . . 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 18820 | . 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 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 SubGrpcsubg 18275 Cntzccntz 18447 Abelcabl 18909 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LVecclvec 19876 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 |
This theorem is referenced by: baerlem3lem1 38845 baerlem5alem1 38846 baerlem5blem1 38847 |
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