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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 21683. (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 6858 | . 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 2733 | . 2 β’ (0gβπ) = (0gβπ) | |
9 | eqid 2733 | . 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 6911 | . . 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 5130 | . 2 β’ (π β (π β π β¦ (π΄βπ)) finSupp (0gβπ)) |
15 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 14 | mptscmfsupp0 20402 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 class class class wbr 5106 β¦ cmpt 5189 βΆwf 6493 βcfv 6497 (class class class)co 7358 finSupp cfsupp 9308 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 LModclmod 20336 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-supp 8094 df-1o 8413 df-en 8887 df-fin 8890 df-fsupp 9309 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-ring 19971 df-lmod 20338 |
This theorem is referenced by: ply1coefsupp 21682 |
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