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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 | β’ π΅ = (Baseβπ) |
suppmptcfin.r | β’ π = (Scalarβπ) |
suppmptcfin.0 | β’ 0 = (0gβπ ) |
suppmptcfin.1 | β’ 1 = (1rβπ ) |
suppmptcfin.f | β’ πΉ = (π₯ β π β¦ if(π₯ = π, 1 , 0 )) |
Ref | Expression |
---|---|
mptcfsupp | β’ ((π β LMod β§ π β π« π΅ β§ π β π) β πΉ finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppmptcfin.f | . . . 4 β’ πΉ = (π₯ β π β¦ if(π₯ = π, 1 , 0 )) | |
2 | 1 | funmpt2 6544 | . . 3 β’ Fun πΉ |
3 | 2 | a1i 11 | . 2 β’ ((π β LMod β§ π β π« π΅ β§ π β π) β Fun πΉ) |
4 | suppmptcfin.b | . . 3 β’ π΅ = (Baseβπ) | |
5 | suppmptcfin.r | . . 3 β’ π = (Scalarβπ) | |
6 | suppmptcfin.0 | . . 3 β’ 0 = (0gβπ ) | |
7 | suppmptcfin.1 | . . 3 β’ 1 = (1rβπ ) | |
8 | 4, 5, 6, 7, 1 | suppmptcfin 46545 | . 2 β’ ((π β LMod β§ π β π« π΅ β§ π β π) β (πΉ supp 0 ) β Fin) |
9 | mptexg 7175 | . . . . 5 β’ (π β π« π΅ β (π₯ β π β¦ if(π₯ = π, 1 , 0 )) β V) | |
10 | 1, 9 | eqeltrid 2838 | . . . 4 β’ (π β π« π΅ β πΉ β V) |
11 | 10 | 3ad2ant2 1135 | . . 3 β’ ((π β LMod β§ π β π« π΅ β§ π β π) β πΉ β V) |
12 | 6 | fvexi 6860 | . . 3 β’ 0 β V |
13 | isfsupp 9315 | . . 3 β’ ((πΉ β V β§ 0 β V) β (πΉ finSupp 0 β (Fun πΉ β§ (πΉ supp 0 ) β Fin))) | |
14 | 11, 12, 13 | sylancl 587 | . 2 β’ ((π β LMod β§ π β π« π΅ β§ π β π) β (πΉ finSupp 0 β (Fun πΉ β§ (πΉ supp 0 ) β Fin))) |
15 | 3, 8, 14 | mpbir2and 712 | 1 β’ ((π β LMod β§ π β π« π΅ β§ π β π) β πΉ finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3447 ifcif 4490 π« cpw 4564 class class class wbr 5109 β¦ cmpt 5192 Fun wfun 6494 βcfv 6500 (class class class)co 7361 supp csupp 8096 Fincfn 8889 finSupp cfsupp 9311 Basecbs 17091 Scalarcsca 17144 0gc0g 17329 1rcur 19921 LModclmod 20365 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-supp 8097 df-1o 8416 df-en 8890 df-fin 8893 df-fsupp 9312 |
This theorem is referenced by: lcoss 46607 el0ldep 46637 |
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