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Mirrors > Home > MPE Home > Th. List > smadiadet | Structured version Visualization version Unicode version |
Description: The determinant of a submatrix of a square matrix obtained by removing a row and a column at the same index equals the determinant of the original matrix with the row replaced with 0's and a 1 at the diagonal position. (Contributed by AV, 31-Jan-2019.) (Proof shortened by AV, 24-Jul-2019.) |
Ref | Expression |
---|---|
smadiadet.a | Mat |
smadiadet.b | |
smadiadet.r | |
smadiadet.d | maDet |
smadiadet.h | maDet |
Ref | Expression |
---|---|
smadiadet | subMat minMatR1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smadiadet.a | . . . . 5 Mat | |
2 | eqid 2622 | . . . . 5 subMat subMat | |
3 | smadiadet.b | . . . . 5 | |
4 | 1, 2, 3 | submaval 20387 | . . . 4 subMat |
5 | 4 | 3anidm23 1385 | . . 3 subMat |
6 | 5 | fveq2d 6195 | . 2 subMat |
7 | eqid 2622 | . . . . . 6 minMatR1 minMatR1 | |
8 | eqid 2622 | . . . . . 6 | |
9 | eqid 2622 | . . . . . 6 | |
10 | 1, 3, 7, 8, 9 | minmar1val 20454 | . . . . 5 minMatR1 |
11 | 10 | 3anidm23 1385 | . . . 4 minMatR1 |
12 | 11 | fveq2d 6195 | . . 3 minMatR1 |
13 | smadiadet.r | . . . . 5 | |
14 | 1, 3, 13, 9, 8 | marep01ma 20466 | . . . . 5 |
15 | smadiadet.d | . . . . . 6 maDet | |
16 | eqid 2622 | . . . . . 6 | |
17 | eqid 2622 | . . . . . 6 RHom RHom | |
18 | eqid 2622 | . . . . . 6 pmSgn pmSgn | |
19 | eqid 2622 | . . . . . 6 | |
20 | eqid 2622 | . . . . . 6 mulGrp mulGrp | |
21 | 15, 1, 3, 16, 17, 18, 19, 20 | mdetleib2 20394 | . . . . 5 g RHom pmSgnmulGrp g |
22 | 13, 14, 21 | sylancr 695 | . . . 4 g RHom pmSgnmulGrp g |
23 | 22 | adantr 481 | . . 3 g RHom pmSgnmulGrp g |
24 | eqid 2622 | . . . . 5 | |
25 | eqid 2622 | . . . . 5 | |
26 | crngring 18558 | . . . . . . 7 | |
27 | ringcmn 18581 | . . . . . . 7 CMnd | |
28 | 13, 26, 27 | mp2b 10 | . . . . . 6 CMnd |
29 | 28 | a1i 11 | . . . . 5 CMnd |
30 | 1, 3 | matrcl 20218 | . . . . . . . 8 |
31 | 30 | simpld 475 | . . . . . . 7 |
32 | eqid 2622 | . . . . . . . 8 | |
33 | 32, 16 | symgbasfi 17806 | . . . . . . 7 |
34 | 31, 33 | syl 17 | . . . . . 6 |
35 | 34 | adantr 481 | . . . . 5 |
36 | 1, 3, 13, 9, 8, 16, 20, 17, 18, 19 | smadiadetlem1 20468 | . . . . 5 RHom pmSgnmulGrp g |
37 | disjdif 4040 | . . . . . 6 | |
38 | 37 | a1i 11 | . . . . 5 |
39 | ssrab2 3687 | . . . . . . . 8 | |
40 | 39 | a1i 11 | . . . . . . 7 |
41 | undif 4049 | . . . . . . 7 | |
42 | 40, 41 | sylib 208 | . . . . . 6 |
43 | 42 | eqcomd 2628 | . . . . 5 |
44 | 24, 25, 29, 35, 36, 38, 43 | gsummptfidmsplit 18330 | . . . 4 g RHom pmSgnmulGrp g g RHom pmSgnmulGrp g g RHom pmSgnmulGrp g |
45 | eqid 2622 | . . . . . 6 | |
46 | eqid 2622 | . . . . . 6 pmSgn pmSgn | |
47 | 1, 3, 13, 9, 8, 16, 20, 17, 18, 19, 45, 46 | smadiadetlem4 20475 | . . . . 5 g RHom pmSgnmulGrp g g RHom pmSgn mulGrp g |
48 | 1, 3, 13, 9, 8, 16, 20, 17, 18, 19 | smadiadetlem2 20470 | . . . . 5 g RHom pmSgnmulGrp g |
49 | 47, 48 | oveq12d 6668 | . . . 4 g RHom pmSgnmulGrp g g RHom pmSgnmulGrp g g RHom pmSgn mulGrp g |
50 | ringmnd 18556 | . . . . . . 7 | |
51 | 13, 26, 50 | mp2b 10 | . . . . . 6 |
52 | diffi 8192 | . . . . . . . . . 10 | |
53 | 31, 52 | syl 17 | . . . . . . . . 9 |
54 | 53 | adantr 481 | . . . . . . . 8 |
55 | eqid 2622 | . . . . . . . . 9 | |
56 | 55, 45 | symgbasfi 17806 | . . . . . . . 8 |
57 | 54, 56 | syl 17 | . . . . . . 7 |
58 | 13 | a1i 11 | . . . . . . . . 9 |
59 | simpll 790 | . . . . . . . . . 10 | |
60 | difssd 3738 | . . . . . . . . . 10 | |
61 | 1, 3 | submabas 20384 | . . . . . . . . . 10 Mat |
62 | 59, 60, 61 | syl2anc 693 | . . . . . . . . 9 Mat |
63 | simpr 477 | . . . . . . . . 9 | |
64 | eqid 2622 | . . . . . . . . . 10 Mat Mat | |
65 | eqid 2622 | . . . . . . . . . 10 Mat Mat | |
66 | 45, 46, 17, 64, 65, 20 | madetsmelbas2 20271 | . . . . . . . . 9 Mat RHom pmSgn mulGrp g |
67 | 58, 62, 63, 66 | syl3anc 1326 | . . . . . . . 8 RHom pmSgn mulGrp g |
68 | 67 | ralrimiva 2966 | . . . . . . 7 RHom pmSgn mulGrp g |
69 | 24, 29, 57, 68 | gsummptcl 18366 | . . . . . 6 g RHom pmSgn mulGrp g |
70 | 24, 25, 9 | mndrid 17312 | . . . . . 6 g RHom pmSgn mulGrp g g RHom pmSgn mulGrp g g RHom pmSgn mulGrp g |
71 | 51, 69, 70 | sylancr 695 | . . . . 5 g RHom pmSgn mulGrp g g RHom pmSgn mulGrp g |
72 | difssd 3738 | . . . . . . 7 | |
73 | 61, 13 | jctil 560 | . . . . . . 7 Mat |
74 | 72, 73 | sylan2 491 | . . . . . 6 Mat |
75 | smadiadet.h | . . . . . . 7 maDet | |
76 | 75, 64, 65, 45, 17, 46, 19, 20 | mdetleib2 20394 | . . . . . 6 Mat g RHom pmSgn mulGrp g |
77 | 74, 76 | syl 17 | . . . . 5 g RHom pmSgn mulGrp g |
78 | 71, 77 | eqtr4d 2659 | . . . 4 g RHom pmSgn mulGrp g |
79 | 44, 49, 78 | 3eqtrd 2660 | . . 3 g RHom pmSgnmulGrp g |
80 | 12, 23, 79 | 3eqtrd 2660 | . 2 minMatR1 |
81 | 6, 80 | eqtr4d 2659 | 1 subMat minMatR1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 cdif 3571 cun 3572 cin 3573 wss 3574 c0 3915 cif 4086 csn 4177 cmpt 4729 ccom 5118 cfv 5888 (class class class)co 6650 cmpt2 6652 cfn 7955 cbs 15857 cplusg 15941 cmulr 15942 c0g 16100 g cgsu 16101 cmnd 17294 csymg 17797 pmSgncpsgn 17909 CMndccmn 18193 mulGrpcmgp 18489 cur 18501 crg 18547 ccrg 18548 RHomczrh 19848 Mat cmat 20213 subMat csubma 20382 maDet cmdat 20390 minMatR1 cminmar1 20439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-sra 19172 df-rgmod 19173 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-dsmm 20076 df-frlm 20091 df-mat 20214 df-subma 20383 df-mdet 20391 df-minmar1 20441 |
This theorem is referenced by: smadiadetg 20479 |
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