Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mptcfsupp | Structured version Visualization version GIF version |
Description: A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
suppmptcfin.b | ⊢ 𝐵 = (Base‘𝑀) |
suppmptcfin.r | ⊢ 𝑅 = (Scalar‘𝑀) |
suppmptcfin.0 | ⊢ 0 = (0g‘𝑅) |
suppmptcfin.1 | ⊢ 1 = (1r‘𝑅) |
suppmptcfin.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) |
Ref | Expression |
---|---|
mptcfsupp | ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppmptcfin.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) | |
2 | 1 | funmpt2 6388 | . . 3 ⊢ Fun 𝐹 |
3 | 2 | a1i 11 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → Fun 𝐹) |
4 | suppmptcfin.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
5 | suppmptcfin.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑀) | |
6 | suppmptcfin.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
7 | suppmptcfin.1 | . . 3 ⊢ 1 = (1r‘𝑅) | |
8 | 4, 5, 6, 7, 1 | suppmptcfin 44421 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin) |
9 | mptexg 6978 | . . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) ∈ V) | |
10 | 1, 9 | eqeltrid 2917 | . . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝐹 ∈ V) |
11 | 10 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V) |
12 | 6 | fvexi 6678 | . . 3 ⊢ 0 ∈ V |
13 | isfsupp 8831 | . . 3 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))) | |
14 | 11, 12, 13 | sylancl 588 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))) |
15 | 3, 8, 14 | mpbir2and 711 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ifcif 4466 𝒫 cpw 4538 class class class wbr 5058 ↦ cmpt 5138 Fun wfun 6343 ‘cfv 6349 (class class class)co 7150 supp csupp 7824 Fincfn 8503 finSupp cfsupp 8827 Basecbs 16477 Scalarcsca 16562 0gc0g 16707 1rcur 19245 LModclmod 19628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-supp 7825 df-1o 8096 df-er 8283 df-en 8504 df-fin 8507 df-fsupp 8828 |
This theorem is referenced by: lcoss 44485 el0ldep 44515 |
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