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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 19099 | . . . . 5 ⊢ (𝑀 ∈ LMod → 0 ∈ 𝑅) |
5 | 4 | ad2antrr 705 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑅) |
6 | lincvalsc0.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) | |
7 | 5, 6 | fmptd 6527 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶𝑅) |
8 | fvex 6342 | . . . . . 6 ⊢ (Base‘𝑆) ∈ V | |
9 | 2, 8 | eqeltri 2846 | . . . . 5 ⊢ 𝑅 ∈ V |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ V) |
11 | elmapg 8022 | . . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶𝑅)) | |
12 | 10, 11 | sylan 569 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶𝑅)) |
13 | 7, 12 | mpbird 247 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ (𝑅 ↑𝑚 𝑉)) |
14 | eqidd 2772 | . . . . . . 7 ⊢ (𝑥 = 𝑣 → 0 = 0 ) | |
15 | 14 | cbvmptv 4884 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ 0 ) = (𝑣 ∈ 𝑉 ↦ 0 ) |
16 | 6, 15 | eqtri 2793 | . . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ 0 ) |
17 | simpr 471 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 𝐵) | |
18 | fvex 6342 | . . . . . . 7 ⊢ (0g‘𝑆) ∈ V | |
19 | 3, 18 | eqeltri 2846 | . . . . . 6 ⊢ 0 ∈ V |
20 | 19 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 0 ∈ V) |
21 | 19 | a1i 11 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑣 ∈ 𝑉) → 0 ∈ V) |
22 | 16, 17, 20, 21 | mptsuppd 7469 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 }) |
23 | neirr 2952 | . . . . . . . 8 ⊢ ¬ 0 ≠ 0 | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ¬ 0 ≠ 0 ) |
25 | 24 | ralrimivw 3116 | . . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) |
26 | rabeq0 4103 | . . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) | |
27 | 25, 26 | sylibr 224 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅) |
28 | 0fin 8344 | . . . . . 6 ⊢ ∅ ∈ Fin | |
29 | 28 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∅ ∈ Fin) |
30 | 27, 29 | eqeltrd 2850 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ∈ Fin) |
31 | 22, 30 | eqeltrd 2850 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) ∈ Fin) |
32 | 6 | funmpt2 6070 | . . . . 5 ⊢ Fun 𝐹 |
33 | 32 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → Fun 𝐹) |
34 | funisfsupp 8436 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (𝑅 ↑𝑚 𝑉) ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) | |
35 | 33, 13, 20, 34 | syl3anc 1476 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) |
36 | 31, 35 | mpbird 247 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 finSupp 0 ) |
37 | lincvalsc0.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
38 | lincvalsc0.z | . . 3 ⊢ 𝑍 = (0g‘𝑀) | |
39 | 37, 1, 3, 38, 6 | lincvalsc0 42738 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) |
40 | 13, 36, 39 | 3jca 1122 | 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 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∀wral 3061 {crab 3065 Vcvv 3351 ∅c0 4063 𝒫 cpw 4297 class class class wbr 4786 ↦ cmpt 4863 Fun wfun 6025 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 supp csupp 7446 ↑𝑚 cmap 8009 Fincfn 8109 finSupp cfsupp 8431 Basecbs 16064 Scalarcsca 16152 0gc0g 16308 LModclmod 19073 linC clinc 42721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-map 8011 df-en 8110 df-fin 8113 df-fsupp 8432 df-seq 13009 df-0g 16310 df-gsum 16311 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-ring 18757 df-lmod 19075 df-linc 42723 |
This theorem is referenced by: lcoel0 42745 |
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