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Mirrors > Home > MPE Home > Th. List > mptscmfsuppd | Structured version Visualization version GIF version |
Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 22221. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.) |
Ref | Expression |
---|---|
mptscmfsuppd.b | β’ π΅ = (Baseβπ) |
mptscmfsuppd.s | β’ π = (Scalarβπ) |
mptscmfsuppd.n | β’ Β· = ( Β·π βπ) |
mptscmfsuppd.p | β’ (π β π β LMod) |
mptscmfsuppd.x | β’ (π β π β π) |
mptscmfsuppd.z | β’ ((π β§ π β π) β π β π΅) |
mptscmfsuppd.a | β’ (π β π΄:πβΆπ) |
mptscmfsuppd.f | β’ (π β π΄ finSupp (0gβπ)) |
Ref | Expression |
---|---|
mptscmfsuppd | β’ (π β (π β π β¦ ((π΄βπ) Β· π)) finSupp (0gβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptscmfsuppd.x | . 2 β’ (π β π β π) | |
2 | mptscmfsuppd.p | . 2 β’ (π β π β LMod) | |
3 | mptscmfsuppd.s | . . 3 β’ π = (Scalarβπ) | |
4 | 3 | a1i 11 | . 2 β’ (π β π = (Scalarβπ)) |
5 | mptscmfsuppd.b | . 2 β’ π΅ = (Baseβπ) | |
6 | fvexd 6905 | . 2 β’ ((π β§ π β π) β (π΄βπ) β V) | |
7 | mptscmfsuppd.z | . 2 β’ ((π β§ π β π) β π β π΅) | |
8 | eqid 2725 | . 2 β’ (0gβπ) = (0gβπ) | |
9 | eqid 2725 | . 2 β’ (0gβπ) = (0gβπ) | |
10 | mptscmfsuppd.n | . 2 β’ Β· = ( Β·π βπ) | |
11 | mptscmfsuppd.a | . . . 4 β’ (π β π΄:πβΆπ) | |
12 | 11 | feqmptd 6960 | . . 3 β’ (π β π΄ = (π β π β¦ (π΄βπ))) |
13 | mptscmfsuppd.f | . . 3 β’ (π β π΄ finSupp (0gβπ)) | |
14 | 12, 13 | eqbrtrrd 5168 | . 2 β’ (π β (π β π β¦ (π΄βπ)) finSupp (0gβπ)) |
15 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 14 | mptscmfsupp0 20809 | 1 β’ (π β (π β π β¦ ((π΄βπ) Β· π)) finSupp (0gβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 class class class wbr 5144 β¦ cmpt 5227 βΆwf 6539 βcfv 6543 (class class class)co 7413 finSupp cfsupp 9380 Basecbs 17174 Scalarcsca 17230 Β·π cvsca 17231 0gc0g 17415 LModclmod 20742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-supp 8159 df-1o 8480 df-en 8958 df-fin 8961 df-fsupp 9381 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18892 df-ring 20174 df-lmod 20744 |
This theorem is referenced by: ply1coefsupp 22220 |
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