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Mirrors > Home > MPE Home > Th. List > smadiadetg0 | Structured version Visualization version GIF version |
Description: Lemma for smadiadetr 21970: version of smadiadetg 21968 with all hypotheses defining class variables removed, i.e. all class variables defined in the hypotheses replaced in the theorem by their definition. (Contributed by AV, 15-Feb-2019.) |
Ref | Expression |
---|---|
smadiadetg0.r | ⊢ 𝑅 ∈ CRing |
Ref | Expression |
---|---|
smadiadetg0 | ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . 2 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
2 | eqid 2736 | . 2 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
3 | smadiadetg0.r | . 2 ⊢ 𝑅 ∈ CRing | |
4 | eqid 2736 | . 2 ⊢ (𝑁 maDet 𝑅) = (𝑁 maDet 𝑅) | |
5 | eqid 2736 | . 2 ⊢ ((𝑁 ∖ {𝐾}) maDet 𝑅) = ((𝑁 ∖ {𝐾}) maDet 𝑅) | |
6 | eqid 2736 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | smadiadetg 21968 | 1 ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∖ cdif 3905 {csn 4584 ‘cfv 6493 (class class class)co 7351 Basecbs 17037 .rcmulr 17088 CRingccrg 19913 Mat cmat 21700 matRRep cmarrep 21851 subMat csubma 21871 maDet cmdat 21879 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-xnn0 12444 df-z 12458 df-dec 12577 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-hash 14185 df-word 14357 df-lsw 14405 df-concat 14413 df-s1 14438 df-substr 14483 df-pfx 14513 df-splice 14592 df-reverse 14601 df-s2 14691 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-starv 17102 df-sca 17103 df-vsca 17104 df-ip 17105 df-tset 17106 df-ple 17107 df-ds 17109 df-unif 17110 df-hom 17111 df-cco 17112 df-0g 17277 df-gsum 17278 df-prds 17283 df-pws 17285 df-mre 17420 df-mrc 17421 df-acs 17423 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-mhm 18555 df-submnd 18556 df-efmnd 18633 df-grp 18705 df-minusg 18706 df-mulg 18826 df-subg 18878 df-ghm 18959 df-gim 19002 df-cntz 19050 df-oppg 19077 df-symg 19102 df-pmtr 19177 df-psgn 19226 df-cmn 19517 df-abl 19518 df-mgp 19850 df-ur 19867 df-ring 19914 df-cring 19915 df-oppr 19996 df-dvdsr 20017 df-unit 20018 df-invr 20048 df-dvr 20059 df-rnghom 20093 df-drng 20134 df-subrg 20167 df-sra 20580 df-rgmod 20581 df-cnfld 20744 df-zring 20817 df-zrh 20851 df-dsmm 21085 df-frlm 21100 df-mat 21701 df-marrep 21853 df-subma 21872 df-mdet 21880 df-minmar1 21930 |
This theorem is referenced by: smadiadetr 21970 |
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