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 5841 | . . 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 41954 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin) |
9 | mptexg 6389 | . . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) ∈ V) | |
10 | 1, 9 | syl5eqel 2692 | . . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝐹 ∈ V) |
11 | 10 | 3ad2ant2 1076 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V) |
12 | fvex 6113 | . . . 4 ⊢ (0g‘𝑅) ∈ V | |
13 | 6, 12 | eqeltri 2684 | . . 3 ⊢ 0 ∈ V |
14 | isfsupp 8162 | . . 3 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))) | |
15 | 11, 13, 14 | sylancl 693 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 finSupp 0 ↔ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))) |
16 | 3, 8, 15 | mpbir2and 959 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 Fincfn 7841 finSupp cfsupp 8158 Basecbs 15695 Scalarcsca 15771 0gc0g 15923 1rcur 18324 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-supp 7183 df-1o 7447 df-er 7629 df-en 7842 df-fin 7845 df-fsupp 8159 |
This theorem is referenced by: lcoss 42019 el0ldep 42049 |
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