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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcoc0 | Structured version Visualization version GIF version | ||
| Description: Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.) | 
| Ref | Expression | 
|---|---|
| lincvalsc0.b | ⊢ 𝐵 = (Base‘𝑀) | 
| lincvalsc0.s | ⊢ 𝑆 = (Scalar‘𝑀) | 
| lincvalsc0.0 | ⊢ 0 = (0g‘𝑆) | 
| lincvalsc0.z | ⊢ 𝑍 = (0g‘𝑀) | 
| lincvalsc0.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) | 
| lcoc0.r | ⊢ 𝑅 = (Base‘𝑆) | 
| Ref | Expression | 
|---|---|
| lcoc0 | ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑉) = 𝑍)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lincvalsc0.s | . . . . . 6 ⊢ 𝑆 = (Scalar‘𝑀) | |
| 2 | lcoc0.r | . . . . . 6 ⊢ 𝑅 = (Base‘𝑆) | |
| 3 | lincvalsc0.0 | . . . . . 6 ⊢ 0 = (0g‘𝑆) | |
| 4 | 1, 2, 3 | lmod0cl 20887 | . . . . 5 ⊢ (𝑀 ∈ LMod → 0 ∈ 𝑅) | 
| 5 | 4 | ad2antrr 726 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑅) | 
| 6 | lincvalsc0.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) | |
| 7 | 5, 6 | fmptd 7133 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶𝑅) | 
| 8 | 2 | fvexi 6919 | . . . . 5 ⊢ 𝑅 ∈ V | 
| 9 | 8 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ V) | 
| 10 | elmapg 8880 | . . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ↔ 𝐹:𝑉⟶𝑅)) | |
| 11 | 9, 10 | sylan 580 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ↔ 𝐹:𝑉⟶𝑅)) | 
| 12 | 7, 11 | mpbird 257 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ (𝑅 ↑m 𝑉)) | 
| 13 | eqidd 2737 | . . . . . . 7 ⊢ (𝑥 = 𝑣 → 0 = 0 ) | |
| 14 | 13 | cbvmptv 5254 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ 0 ) = (𝑣 ∈ 𝑉 ↦ 0 ) | 
| 15 | 6, 14 | eqtri 2764 | . . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ 0 ) | 
| 16 | simpr 484 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 𝐵) | |
| 17 | 3 | fvexi 6919 | . . . . . 6 ⊢ 0 ∈ V | 
| 18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 0 ∈ V) | 
| 19 | 17 | a1i 11 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 0 ∈ V) | 
| 20 | 15, 16, 18, 19 | mptsuppd 8213 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 }) | 
| 21 | neirr 2948 | . . . . . . . 8 ⊢ ¬ 0 ≠ 0 | |
| 22 | 21 | a1i 11 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ¬ 0 ≠ 0 ) | 
| 23 | 22 | ralrimivw 3149 | . . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) | 
| 24 | rabeq0 4387 | . . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) | |
| 25 | 23, 24 | sylibr 234 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅) | 
| 26 | 0fi 9083 | . . . . . 6 ⊢ ∅ ∈ Fin | |
| 27 | 26 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∅ ∈ Fin) | 
| 28 | 25, 27 | eqeltrd 2840 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ∈ Fin) | 
| 29 | 20, 28 | eqeltrd 2840 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) ∈ Fin) | 
| 30 | 6 | funmpt2 6604 | . . . . 5 ⊢ Fun 𝐹 | 
| 31 | 30 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → Fun 𝐹) | 
| 32 | funisfsupp 9408 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (𝑅 ↑m 𝑉) ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) | |
| 33 | 31, 12, 18, 32 | syl3anc 1372 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) | 
| 34 | 29, 33 | mpbird 257 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 finSupp 0 ) | 
| 35 | lincvalsc0.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
| 36 | lincvalsc0.z | . . 3 ⊢ 𝑍 = (0g‘𝑀) | |
| 37 | 35, 1, 3, 36, 6 | lincvalsc0 48343 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) | 
| 38 | 12, 34, 37 | 3jca 1128 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑉) = 𝑍)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 {crab 3435 Vcvv 3479 ∅c0 4332 𝒫 cpw 4599 class class class wbr 5142 ↦ cmpt 5224 Fun wfun 6554 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 supp csupp 8186 ↑m cmap 8867 Fincfn 8986 finSupp cfsupp 9402 Basecbs 17248 Scalarcsca 17301 0gc0g 17485 LModclmod 20859 linC clinc 48326 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-map 8869 df-en 8987 df-fin 8990 df-fsupp 9403 df-seq 14044 df-0g 17487 df-gsum 17488 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-ring 20233 df-lmod 20861 df-linc 48328 | 
| This theorem is referenced by: lcoel0 48350 | 
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