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Mirrors > Home > MPE Home > Th. List > dsmm0cl | Structured version Visualization version GIF version |
Description: The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
Ref | Expression |
---|---|
dsmmcl.p | β’ π = (πXsπ ) |
dsmmcl.h | β’ π» = (Baseβ(π βm π )) |
dsmmcl.i | β’ (π β πΌ β π) |
dsmmcl.s | β’ (π β π β π) |
dsmmcl.r | β’ (π β π :πΌβΆMnd) |
dsmm0cl.z | β’ 0 = (0gβπ) |
Ref | Expression |
---|---|
dsmm0cl | β’ (π β 0 β π») |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dsmmcl.p | . . . 4 β’ π = (πXsπ ) | |
2 | dsmmcl.i | . . . 4 β’ (π β πΌ β π) | |
3 | dsmmcl.s | . . . 4 β’ (π β π β π) | |
4 | dsmmcl.r | . . . 4 β’ (π β π :πΌβΆMnd) | |
5 | 1, 2, 3, 4 | prdsmndd 18657 | . . 3 β’ (π β π β Mnd) |
6 | eqid 2732 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
7 | dsmm0cl.z | . . . 4 β’ 0 = (0gβπ) | |
8 | 6, 7 | mndidcl 18639 | . . 3 β’ (π β Mnd β 0 β (Baseβπ)) |
9 | 5, 8 | syl 17 | . 2 β’ (π β 0 β (Baseβπ)) |
10 | 1, 2, 3, 4 | prds0g 18658 | . . . . . . . . . 10 β’ (π β (0g β π ) = (0gβπ)) |
11 | 10, 7 | eqtr4di 2790 | . . . . . . . . 9 β’ (π β (0g β π ) = 0 ) |
12 | 11 | adantr 481 | . . . . . . . 8 β’ ((π β§ π β πΌ) β (0g β π ) = 0 ) |
13 | 12 | fveq1d 6893 | . . . . . . 7 β’ ((π β§ π β πΌ) β ((0g β π )βπ) = ( 0 βπ)) |
14 | 4 | ffnd 6718 | . . . . . . . 8 β’ (π β π Fn πΌ) |
15 | fvco2 6988 | . . . . . . . 8 β’ ((π Fn πΌ β§ π β πΌ) β ((0g β π )βπ) = (0gβ(π βπ))) | |
16 | 14, 15 | sylan 580 | . . . . . . 7 β’ ((π β§ π β πΌ) β ((0g β π )βπ) = (0gβ(π βπ))) |
17 | 13, 16 | eqtr3d 2774 | . . . . . 6 β’ ((π β§ π β πΌ) β ( 0 βπ) = (0gβ(π βπ))) |
18 | nne 2944 | . . . . . 6 β’ (Β¬ ( 0 βπ) β (0gβ(π βπ)) β ( 0 βπ) = (0gβ(π βπ))) | |
19 | 17, 18 | sylibr 233 | . . . . 5 β’ ((π β§ π β πΌ) β Β¬ ( 0 βπ) β (0gβ(π βπ))) |
20 | 19 | ralrimiva 3146 | . . . 4 β’ (π β βπ β πΌ Β¬ ( 0 βπ) β (0gβ(π βπ))) |
21 | rabeq0 4384 | . . . 4 β’ ({π β πΌ β£ ( 0 βπ) β (0gβ(π βπ))} = β β βπ β πΌ Β¬ ( 0 βπ) β (0gβ(π βπ))) | |
22 | 20, 21 | sylibr 233 | . . 3 β’ (π β {π β πΌ β£ ( 0 βπ) β (0gβ(π βπ))} = β ) |
23 | 0fin 9170 | . . 3 β’ β β Fin | |
24 | 22, 23 | eqeltrdi 2841 | . 2 β’ (π β {π β πΌ β£ ( 0 βπ) β (0gβ(π βπ))} β Fin) |
25 | eqid 2732 | . . 3 β’ (π βm π ) = (π βm π ) | |
26 | dsmmcl.h | . . 3 β’ π» = (Baseβ(π βm π )) | |
27 | 1, 25, 6, 26, 2, 14 | dsmmelbas 21293 | . 2 β’ (π β ( 0 β π» β ( 0 β (Baseβπ) β§ {π β πΌ β£ ( 0 βπ) β (0gβ(π βπ))} β Fin))) |
28 | 9, 24, 27 | mpbir2and 711 | 1 β’ (π β 0 β π») |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 {crab 3432 β c0 4322 β ccom 5680 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7408 Fincfn 8938 Basecbs 17143 0gc0g 17384 Xscprds 17390 Mndcmnd 18624 βm cdsmm 21285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-prds 17392 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-dsmm 21286 |
This theorem is referenced by: dsmmsubg 21297 |
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