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Mirrors > Home > MPE Home > Th. List > smadiadetlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for smadiadet 22494: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to itself. (Contributed by AV, 31-Dec-2018.) |
Ref | Expression |
---|---|
marep01ma.a | โข ๐ด = (๐ Mat ๐ ) |
marep01ma.b | โข ๐ต = (Baseโ๐ด) |
marep01ma.r | โข ๐ โ CRing |
marep01ma.0 | โข 0 = (0gโ๐ ) |
marep01ma.1 | โข 1 = (1rโ๐ ) |
smadiadetlem.p | โข ๐ = (Baseโ(SymGrpโ๐)) |
smadiadetlem.g | โข ๐บ = (mulGrpโ๐ ) |
madetminlem.y | โข ๐ = (โคRHomโ๐ ) |
madetminlem.s | โข ๐ = (pmSgnโ๐) |
madetminlem.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
smadiadetlem2 | โข ((๐ โ ๐ต โง ๐พ โ ๐) โ (๐ ฮฃg (๐ โ (๐ โ {๐ โ ๐ โฃ (๐โ๐พ) = ๐พ}) โฆ (((๐ โ ๐)โ๐) ยท (๐บ ฮฃg (๐ โ ๐ โฆ (๐(๐ โ ๐, ๐ โ ๐ โฆ if(๐ = ๐พ, if(๐ = ๐พ, 1 , 0 ), (๐๐๐)))(๐โ๐))))))) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marep01ma.a | . . 3 โข ๐ด = (๐ Mat ๐ ) | |
2 | marep01ma.b | . . 3 โข ๐ต = (Baseโ๐ด) | |
3 | marep01ma.r | . . 3 โข ๐ โ CRing | |
4 | marep01ma.0 | . . 3 โข 0 = (0gโ๐ ) | |
5 | marep01ma.1 | . . 3 โข 1 = (1rโ๐ ) | |
6 | smadiadetlem.p | . . 3 โข ๐ = (Baseโ(SymGrpโ๐)) | |
7 | smadiadetlem.g | . . 3 โข ๐บ = (mulGrpโ๐ ) | |
8 | madetminlem.y | . . 3 โข ๐ = (โคRHomโ๐ ) | |
9 | madetminlem.s | . . 3 โข ๐ = (pmSgnโ๐) | |
10 | madetminlem.t | . . 3 โข ยท = (.rโ๐ ) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | smadiadetlem1a 22487 | . 2 โข ((๐ โ ๐ต โง ๐พ โ ๐ โง ๐พ โ ๐) โ (๐ ฮฃg (๐ โ (๐ โ {๐ โ ๐ โฃ (๐โ๐พ) = ๐พ}) โฆ (((๐ โ ๐)โ๐) ยท (๐บ ฮฃg (๐ โ ๐ โฆ (๐(๐ โ ๐, ๐ โ ๐ โฆ if(๐ = ๐พ, if(๐ = ๐พ, 1 , 0 ), (๐๐๐)))(๐โ๐))))))) = 0 ) |
12 | 11 | 3anidm23 1418 | 1 โข ((๐ โ ๐ต โง ๐พ โ ๐) โ (๐ ฮฃg (๐ โ (๐ โ {๐ โ ๐ โฃ (๐โ๐พ) = ๐พ}) โฆ (((๐ โ ๐)โ๐) ยท (๐บ ฮฃg (๐ โ ๐ โฆ (๐(๐ โ ๐, ๐ โ ๐ โฆ if(๐ = ๐พ, if(๐ = ๐พ, 1 , 0 ), (๐๐๐)))(๐โ๐))))))) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1533 โ wcel 2098 {crab 3424 โ cdif 3937 ifcif 4520 โฆ cmpt 5221 โ ccom 5670 โcfv 6533 (class class class)co 7401 โ cmpo 7403 Basecbs 17143 .rcmulr 17197 0gc0g 17384 ฮฃg cgsu 17385 SymGrpcsymg 19276 pmSgncpsgn 19399 mulGrpcmgp 20029 1rcur 20076 CRingccrg 20129 โคRHomczrh 21354 Mat cmat 22229 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-word 14462 df-lsw 14510 df-concat 14518 df-s1 14543 df-substr 14588 df-pfx 14618 df-splice 14697 df-reverse 14706 df-s2 14796 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-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-efmnd 18784 df-grp 18856 df-minusg 18857 df-mulg 18986 df-subg 19040 df-ghm 19129 df-gim 19174 df-cntz 19223 df-oppg 19252 df-symg 19277 df-pmtr 19352 df-psgn 19401 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20436 df-subrg 20461 df-sra 21011 df-rgmod 21012 df-cnfld 21229 df-zring 21302 df-zrh 21358 df-dsmm 21595 df-frlm 21610 df-mat 22230 |
This theorem is referenced by: smadiadet 22494 |
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