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Mirrors > Home > MPE Home > Th. List > smadiadetlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for smadiadet 21414: 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 | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
marep01ma.b | ⊢ 𝐵 = (Base‘𝐴) |
marep01ma.r | ⊢ 𝑅 ∈ CRing |
marep01ma.0 | ⊢ 0 = (0g‘𝑅) |
marep01ma.1 | ⊢ 1 = (1r‘𝑅) |
smadiadetlem.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
smadiadetlem.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
madetminlem.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
madetminlem.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
madetminlem.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
smadiadetlem2 | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marep01ma.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | marep01ma.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
3 | marep01ma.r | . . 3 ⊢ 𝑅 ∈ CRing | |
4 | marep01ma.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
5 | marep01ma.1 | . . 3 ⊢ 1 = (1r‘𝑅) | |
6 | smadiadetlem.p | . . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
7 | smadiadetlem.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
8 | madetminlem.y | . . 3 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
9 | madetminlem.s | . . 3 ⊢ 𝑆 = (pmSgn‘𝑁) | |
10 | madetminlem.t | . . 3 ⊢ · = (.r‘𝑅) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | smadiadetlem1a 21407 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 ) |
12 | 11 | 3anidm23 1422 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {crab 3057 ∖ cdif 3838 ifcif 4411 ↦ cmpt 5107 ∘ ccom 5523 ‘cfv 6333 (class class class)co 7164 ∈ cmpo 7166 Basecbs 16579 .rcmulr 16662 0gc0g 16809 Σg cgsu 16810 SymGrpcsymg 18606 pmSgncpsgn 18728 mulGrpcmgp 19351 1rcur 19363 CRingccrg 19410 ℤRHomczrh 20313 Mat cmat 21151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-addf 10687 ax-mulf 10688 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-xor 1507 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-ot 4522 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-sup 8972 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-xnn0 12042 df-z 12056 df-dec 12173 df-uz 12318 df-rp 12466 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-hash 13776 df-word 13949 df-lsw 13997 df-concat 14005 df-s1 14032 df-substr 14085 df-pfx 14115 df-splice 14194 df-reverse 14203 df-s2 14292 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-0g 16811 df-gsum 16812 df-prds 16817 df-pws 16819 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-submnd 18066 df-efmnd 18143 df-grp 18215 df-minusg 18216 df-mulg 18336 df-subg 18387 df-ghm 18467 df-gim 18510 df-cntz 18558 df-oppg 18585 df-symg 18607 df-pmtr 18681 df-psgn 18730 df-cmn 19019 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-rnghom 19582 df-subrg 19645 df-sra 20056 df-rgmod 20057 df-cnfld 20211 df-zring 20283 df-zrh 20317 df-dsmm 20541 df-frlm 20556 df-mat 21152 |
This theorem is referenced by: smadiadet 21414 |
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