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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 20277 | . . . . 5 ⊢ (𝑀 ∈ LMod → 0 ∈ 𝑅) |
5 | 4 | ad2antrr 725 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑅) |
6 | lincvalsc0.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) | |
7 | 5, 6 | fmptd 7057 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶𝑅) |
8 | 2 | fvexi 6852 | . . . . 5 ⊢ 𝑅 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ V) |
10 | elmapg 8712 | . . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ↔ 𝐹:𝑉⟶𝑅)) | |
11 | 9, 10 | sylan 581 | . . 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 2739 | . . . . . . 7 ⊢ (𝑥 = 𝑣 → 0 = 0 ) | |
14 | 13 | cbvmptv 5217 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ 0 ) = (𝑣 ∈ 𝑉 ↦ 0 ) |
15 | 6, 14 | eqtri 2766 | . . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ 0 ) |
16 | simpr 486 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 𝐵) | |
17 | 3 | fvexi 6852 | . . . . . 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 8086 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 }) |
21 | neirr 2951 | . . . . . . . 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 4343 | . . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) | |
25 | 23, 24 | sylibr 233 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅) |
26 | 0fin 9049 | . . . . . 6 ⊢ ∅ ∈ Fin | |
27 | 26 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∅ ∈ Fin) |
28 | 25, 27 | eqeltrd 2839 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ∈ Fin) |
29 | 20, 28 | eqeltrd 2839 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) ∈ Fin) |
30 | 6 | funmpt2 6536 | . . . . 5 ⊢ Fun 𝐹 |
31 | 30 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → Fun 𝐹) |
32 | funisfsupp 9244 | . . . 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 46293 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) |
38 | 12, 34, 37 | 3jca 1129 | 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 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∀wral 3063 {crab 3406 Vcvv 3444 ∅c0 4281 𝒫 cpw 4559 class class class wbr 5104 ↦ cmpt 5187 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7350 supp csupp 8060 ↑m cmap 8699 Fincfn 8817 finSupp cfsupp 9239 Basecbs 17019 Scalarcsca 17072 0gc0g 17257 LModclmod 20251 linC clinc 46276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-map 8701 df-en 8818 df-fin 8821 df-fsupp 9240 df-seq 13837 df-0g 17259 df-gsum 17260 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-grp 18687 df-ring 19896 df-lmod 20253 df-linc 46278 |
This theorem is referenced by: lcoel0 46300 |
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