Theorem smadiadetr 22640

Description: The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. Closed form of smadiadetg 22638. Special case of the "Laplace expansion", see definition in [Lang] p. 515. (Contributed by AV, 15-Feb-2019.)

Assertion

Ref	Expression
smadiadetr	⊢ (((𝑅 ∈ CRing ∧ 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))))

Proof of Theorem smadiadetr

Step	Hyp	Ref	Expression
1		3anass 1095	. . . . 5 ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ↔ (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))))
2		oveq2 7375	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 Mat 𝑅) = (𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
3	2	fveq2d 6844	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))))
4	3	eleq2d 2822	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ↔ 𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
5		fveq2 6840	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘𝑅) = (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
6	5	eleq2d 2822	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑆 ∈ (Base‘𝑅) ↔ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))))
7	4, 6	3anbi13d 1441	. . . . 5 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ↔ (𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
8	1, 7	bitr3id 285	. . . 4 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) ↔ (𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
9		oveq2 7375	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 maDet 𝑅) = (𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
10		oveq2 7375	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 matRRep 𝑅) = (𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld)))
11	10	oveqd 7384	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀(𝑁 matRRep 𝑅)𝑆) = (𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆))
12	11	oveqd 7384	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾))
13	9, 12	fveq12d 6847	. . . . 5 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)))
14		fveq2 6840	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (.r‘𝑅) = (.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
15		eqidd 2737	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → 𝑆 = 𝑆)
16		oveq2 7375	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 ∖ {𝐾}) maDet 𝑅) = ((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
17		oveq2 7375	. . . . . . . . 9 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 subMat 𝑅) = (𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
18	17	fveq1d 6842	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 subMat 𝑅)‘𝑀) = ((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀))
19	18	oveqd 7384	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾) = (𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))
20	16, 19	fveq12d 6847	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾)))
21	14, 15, 20	oveq123d 7388	. . . . 5 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))))
22	13, 21	eqeq12d 2752	. . . 4 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))) ↔ ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾)))))
23	8, 22	imbi12d 344	. . 3 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (((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)))) ↔ ((𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))) → ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))))))
24		cncrng 21373	. . . . 5 ⊢ ℂfld ∈ CRing
25	24	elimel 4536	. . . 4 ⊢ if(𝑅 ∈ CRing, 𝑅, ℂfld) ∈ CRing
26	25	smadiadetg0 22639	. . 3 ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))) → ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))))
27	23, 26	dedth 4525	. 2 ⊢ (𝑅 ∈ CRing → ((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)))))
28	27	impl 455	1 ⊢ (((𝑅 ∈ CRing ∧ 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))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886  ifcif 4466  {csn 4567  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  ._rcmulr 17221  CRingccrg 20215  ℂ_fldccnfld 21352   Mat cmat 22372   matRRep cmarrep 22521   subMat csubma 22541   maDet cmdat 22549

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-addf 11117  ax-mulf 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-splice 14712  df-reverse 14721  df-s2 14810  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-efmnd 18837  df-grp 18912  df-minusg 18913  df-mulg 19044  df-subg 19099  df-ghm 19188  df-gim 19234  df-cntz 19292  df-oppg 19321  df-symg 19345  df-pmtr 19417  df-psgn 19466  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-dvr 20381  df-rhm 20452  df-subrng 20523  df-subrg 20547  df-drng 20708  df-sra 21168  df-rgmod 21169  df-cnfld 21353  df-zring 21427  df-zrh 21483  df-dsmm 21712  df-frlm 21727  df-mat 22373  df-marrep 22523  df-subma 22542  df-mdet 22550  df-minmar1 22600

This theorem is referenced by:  cramerimplem1  22648  madjusmdetlem1  33971