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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 19663 | . . . . 5 ⊢ (𝑀 ∈ LMod → 0 ∈ 𝑅) |
5 | 4 | ad2antrr 724 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑅) |
6 | lincvalsc0.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ 0 ) | |
7 | 5, 6 | fmptd 6881 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹:𝑉⟶𝑅) |
8 | 2 | fvexi 6687 | . . . . 5 ⊢ 𝑅 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ V) |
10 | elmapg 8422 | . . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ↔ 𝐹:𝑉⟶𝑅)) | |
11 | 9, 10 | sylan 582 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅 ↑m 𝑉) ↔ 𝐹:𝑉⟶𝑅)) |
12 | 7, 11 | mpbird 259 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝐹 ∈ (𝑅 ↑m 𝑉)) |
13 | eqidd 2825 | . . . . . . 7 ⊢ (𝑥 = 𝑣 → 0 = 0 ) | |
14 | 13 | cbvmptv 5172 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ 0 ) = (𝑣 ∈ 𝑉 ↦ 0 ) |
15 | 6, 14 | eqtri 2847 | . . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ 0 ) |
16 | simpr 487 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 𝐵) | |
17 | 3 | fvexi 6687 | . . . . . 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 7856 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 }) |
21 | neirr 3028 | . . . . . . . 8 ⊢ ¬ 0 ≠ 0 | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ¬ 0 ≠ 0 ) |
23 | 22 | ralrimivw 3186 | . . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) |
24 | rabeq0 4341 | . . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) | |
25 | 23, 24 | sylibr 236 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅) |
26 | 0fin 8749 | . . . . . 6 ⊢ ∅ ∈ Fin | |
27 | 26 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ∅ ∈ Fin) |
28 | 25, 27 | eqeltrd 2916 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → {𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ∈ Fin) |
29 | 20, 28 | eqeltrd 2916 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 supp 0 ) ∈ Fin) |
30 | 6 | funmpt2 6397 | . . . . 5 ⊢ Fun 𝐹 |
31 | 30 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → Fun 𝐹) |
32 | funisfsupp 8841 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (𝑅 ↑m 𝑉) ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) | |
33 | 31, 12, 18, 32 | syl3anc 1367 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 finSupp 0 ↔ (𝐹 supp 0 ) ∈ Fin)) |
34 | 29, 33 | mpbird 259 | . 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 44483 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍) |
38 | 12, 34, 37 | 3jca 1124 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 {crab 3145 Vcvv 3497 ∅c0 4294 𝒫 cpw 4542 class class class wbr 5069 ↦ cmpt 5149 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 supp csupp 7833 ↑m cmap 8409 Fincfn 8512 finSupp cfsupp 8836 Basecbs 16486 Scalarcsca 16571 0gc0g 16716 LModclmod 19637 linC clinc 44466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-map 8411 df-en 8513 df-fin 8516 df-fsupp 8837 df-seq 13373 df-0g 16718 df-gsum 16719 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-ring 19302 df-lmod 19639 df-linc 44468 |
This theorem is referenced by: lcoel0 44490 |
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