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 6397 | . . 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 44434 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin) |
9 | mptexg 6987 | . . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) ∈ V) | |
10 | 1, 9 | eqeltrid 2920 | . . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝐹 ∈ V) |
11 | 10 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V) |
12 | 6 | fvexi 6687 | . . 3 ⊢ 0 ∈ V |
13 | isfsupp 8840 | . . 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 1536 ∈ wcel 2113 Vcvv 3497 ifcif 4470 𝒫 cpw 4542 class class class wbr 5069 ↦ cmpt 5149 Fun wfun 6352 ‘cfv 6358 (class class class)co 7159 supp csupp 7833 Fincfn 8512 finSupp cfsupp 8836 Basecbs 16486 Scalarcsca 16571 0gc0g 16716 1rcur 19254 LModclmod 19637 |
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-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-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-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-supp 7834 df-1o 8105 df-er 8292 df-en 8513 df-fin 8516 df-fsupp 8837 |
This theorem is referenced by: lcoss 44498 el0ldep 44528 |
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