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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 19204 | . . . . 5 ⊢ (𝑀 ∈ LMod → 0 ∈ 𝑅) |
5 | 4 | ad2antrr 718 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑅) |
6 | lincvalsc0.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) | |
7 | 5, 6 | fmptd 6608 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶𝑅) |
8 | 2 | fvexi 6423 | . . . . 5 ⊢ 𝑅 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ V) |
10 | elmapg 8106 | . . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶𝑅)) | |
11 | 9, 10 | sylan 576 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶𝑅)) |
12 | 7, 11 | mpbird 249 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ (𝑅 ↑𝑚 𝑉)) |
13 | eqidd 2798 | . . . . . . 7 ⊢ (𝑥 = 𝑣 → 0 = 0 ) | |
14 | 13 | cbvmptv 4941 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ 0 ) = (𝑣 ∈ 𝑉 ↦ 0 ) |
15 | 6, 14 | eqtri 2819 | . . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ 0 ) |
16 | simpr 478 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 𝐵) | |
17 | 3 | fvexi 6423 | . . . . . 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 7553 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 }) |
21 | neirr 2978 | . . . . . . . 8 ⊢ ¬ 0 ≠ 0 | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ¬ 0 ≠ 0 ) |
23 | 22 | ralrimivw 3146 | . . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) |
24 | rabeq0 4155 | . . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) | |
25 | 23, 24 | sylibr 226 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅) |
26 | 0fin 8428 | . . . . . 6 ⊢ ∅ ∈ Fin | |
27 | 26 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∅ ∈ Fin) |
28 | 25, 27 | eqeltrd 2876 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ∈ Fin) |
29 | 20, 28 | eqeltrd 2876 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) ∈ Fin) |
30 | 6 | funmpt2 6138 | . . . . 5 ⊢ Fun 𝐹 |
31 | 30 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → Fun 𝐹) |
32 | funisfsupp 8520 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (𝑅 ↑𝑚 𝑉) ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) | |
33 | 31, 12, 18, 32 | syl3anc 1491 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) |
34 | 29, 33 | mpbird 249 | . 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 42997 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) |
38 | 12, 34, 37 | 3jca 1159 | 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 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∀wral 3087 {crab 3091 Vcvv 3383 ∅c0 4113 𝒫 cpw 4347 class class class wbr 4841 ↦ cmpt 4920 Fun wfun 6093 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 supp csupp 7530 ↑𝑚 cmap 8093 Fincfn 8193 finSupp cfsupp 8515 Basecbs 16181 Scalarcsca 16267 0gc0g 16412 LModclmod 19178 linC clinc 42980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-map 8095 df-en 8194 df-fin 8197 df-fsupp 8516 df-seq 13052 df-0g 16414 df-gsum 16415 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-ring 18862 df-lmod 19180 df-linc 42982 |
This theorem is referenced by: lcoel0 43004 |
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