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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 18889 | . . . . 5 ⊢ (𝑀 ∈ LMod → 0 ∈ 𝑅) |
5 | 4 | ad2antrr 762 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑅) |
6 | lincvalsc0.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) | |
7 | 5, 6 | fmptd 6385 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶𝑅) |
8 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑆) ∈ V | |
9 | 2, 8 | eqeltri 2697 | . . . . 5 ⊢ 𝑅 ∈ V |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ V) |
11 | elmapg 7870 | . . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶𝑅)) | |
12 | 10, 11 | sylan 488 | . . 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 2623 | . . . . . . 7 ⊢ (𝑥 = 𝑣 → 0 = 0 ) | |
15 | 14 | cbvmptv 4750 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ 0 ) = (𝑣 ∈ 𝑉 ↦ 0 ) |
16 | 6, 15 | eqtri 2644 | . . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ 0 ) |
17 | simpr 477 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 𝐵) | |
18 | fvex 6201 | . . . . . . 7 ⊢ (0g‘𝑆) ∈ V | |
19 | 3, 18 | eqeltri 2697 | . . . . . 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 7318 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 }) |
23 | neirr 2803 | . . . . . . . 8 ⊢ ¬ 0 ≠ 0 | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ¬ 0 ≠ 0 ) |
25 | 24 | ralrimivw 2967 | . . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) |
26 | rabeq0 3957 | . . . . . 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 8188 | . . . . . 6 ⊢ ∅ ∈ Fin | |
29 | 28 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∅ ∈ Fin) |
30 | 27, 29 | eqeltrd 2701 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ∈ Fin) |
31 | 22, 30 | eqeltrd 2701 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) ∈ Fin) |
32 | 6 | funmpt2 5927 | . . . . 5 ⊢ Fun 𝐹 |
33 | 32 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → Fun 𝐹) |
34 | funisfsupp 8280 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (𝑅 ↑𝑚 𝑉) ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) | |
35 | 33, 13, 20, 34 | syl3anc 1326 | . . 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 42210 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) |
40 | 13, 36, 39 | 3jca 1242 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 Vcvv 3200 ∅c0 3915 𝒫 cpw 4158 class class class wbr 4653 ↦ cmpt 4729 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supp csupp 7295 ↑𝑚 cmap 7857 Fincfn 7955 finSupp cfsupp 8275 Basecbs 15857 Scalarcsca 15944 0gc0g 16100 LModclmod 18863 linC clinc 42193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-map 7859 df-en 7956 df-fin 7959 df-fsupp 8276 df-seq 12802 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ring 18549 df-lmod 18865 df-linc 42195 |
This theorem is referenced by: lcoel0 42217 |
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