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Mirrors > Home > MPE Home > Th. List > smadiadetlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for smadiadet 22585: 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 22578 | . 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 394 = wceq 1533 โ wcel 2098 {crab 3419 โ cdif 3938 ifcif 4525 โฆ cmpt 5227 โ ccom 5677 โcfv 6543 (class class class)co 7413 โ cmpo 7415 Basecbs 17174 .rcmulr 17228 0gc0g 17415 ฮฃg cgsu 17416 SymGrpcsymg 19320 pmSgncpsgn 19443 mulGrpcmgp 20073 1rcur 20120 CRingccrg 20173 โคRHomczrh 21424 Mat cmat 22320 |
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-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-addf 11212 ax-mulf 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 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-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 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-se 5629 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-pred 6301 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-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-xnn0 12570 df-z 12584 df-dec 12703 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-word 14492 df-lsw 14540 df-concat 14548 df-s1 14573 df-substr 14618 df-pfx 14648 df-splice 14727 df-reverse 14736 df-s2 14826 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-efmnd 18820 df-grp 18892 df-minusg 18893 df-mulg 19023 df-subg 19077 df-ghm 19167 df-gim 19212 df-cntz 19267 df-oppg 19296 df-symg 19321 df-pmtr 19396 df-psgn 19445 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-sra 21057 df-rgmod 21058 df-cnfld 21279 df-zring 21372 df-zrh 21428 df-dsmm 21665 df-frlm 21680 df-mat 22321 |
This theorem is referenced by: smadiadet 22585 |
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