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Theorem mdetero 22497

Description: The determinant function is multilinear (additive and homogeneous for each row (matrices are given explicitly by their entries). Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 16-Jul-2018.)

**Hypotheses**
Ref	Expression
mdetero.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetero.k	⊢ 𝐾 = (Base‘𝑅)
mdetero.p	⊢ + = (+_g‘𝑅)
mdetero.t	⊢ · = (._r‘𝑅)
mdetero.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mdetero.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetero.x	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
mdetero.y	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
mdetero.z	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)
mdetero.w	⊢ (𝜑 → 𝑊 ∈ 𝐾)
mdetero.i	⊢ (𝜑 → 𝐼 ∈ 𝑁)
mdetero.j	⊢ (𝜑 → 𝐽 ∈ 𝑁)
mdetero.ij	⊢ (𝜑 → 𝐼 ≠ 𝐽)

**Assertion**
Ref	Expression
mdetero	⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))

Distinct variable groups: 𝜑,𝑖,𝑗 𝑖,𝐾,𝑗 𝑖,𝑁,𝑗 𝑖,𝐼,𝑗 𝑖,𝐽,𝑗 𝑖,𝑋 𝑖,𝑌 𝑖,𝑊,𝑗 · ,𝑖,𝑗 + ,𝑖,𝑗 Allowed substitution hints: 𝐷(𝑖,𝑗) 𝑅(𝑖,𝑗) 𝑋(𝑗) 𝑌(𝑗) 𝑍(𝑖,𝑗)

**Proof of Theorem mdetero**
Step	Hyp	Ref	Expression
1		mdetero.d	. . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅)
2		mdetero.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
3		mdetero.p	. . 3 ⊢ + = (+_g‘𝑅)
4		mdetero.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
5		mdetero.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
6		mdetero.x	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
7	6	3adant2 1131	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
8		crngring 20154	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	4, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	9	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
11		mdetero.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐾)
12	11	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑊 ∈ 𝐾)
13		mdetero.y	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
14	13	3adant2 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
15		mdetero.t	. . . . 5 ⊢ · = (._r‘𝑅)
16	2, 15	ringcl 20159	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑊 · 𝑌) ∈ 𝐾)
17	10, 12, 14, 16	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑊 · 𝑌) ∈ 𝐾)
18		mdetero.z	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)
19	14, 18	ifcld 4535	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐽, 𝑌, 𝑍) ∈ 𝐾)
20		mdetero.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑁)
21	1, 2, 3, 4, 5, 7, 17, 19, 20	mdetrlin2 22494	. 2 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍))))))
22	1, 2, 15, 4, 5, 14, 19, 11, 20	mdetrsca2 22491	. . . 4 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝑊 · (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍))))))
23		eqid 2729	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
24		mdetero.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝑁)
25		mdetero.ij	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
26	1, 2, 23, 4, 5, 13, 18, 20, 24, 25	mdetralt2 22496	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
27	26	oveq2d 7403	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍))))) = (𝑊 · (0_g‘𝑅)))
28	2, 15, 23	ringrz 20203	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾) → (𝑊 · (0_g‘𝑅)) = (0_g‘𝑅))
29	9, 11, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑊 · (0_g‘𝑅)) = (0_g‘𝑅))
30	22, 27, 29	3eqtrd 2768	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
31	30	oveq2d 7403	. 2 ⊢ (𝜑 → ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍))))) = ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (0_g‘𝑅)))
32		ringgrp 20147	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
33	9, 32	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
34		eqid 2729	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
35		eqid 2729	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
36	1, 34, 35, 2	mdetf 22482	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾)
37	4, 36	syl 17	. . . 4 ⊢ (𝜑 → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾)
38	7, 19	ifcld 4535	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)) ∈ 𝐾)
39	34, 2, 35, 5, 4, 38	matbas2d 22310	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))) ∈ (Base‘(𝑁 Mat 𝑅)))
40	37, 39	ffvelcdmd 7057	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) ∈ 𝐾)
41	2, 3, 23	grprid 18900	. . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) ∈ 𝐾) → ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (0_g‘𝑅)) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))
42	33, 40, 41	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) + (0_g‘𝑅)) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))
43	21, 31, 42	3eqtrd 2768	1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (𝑊 · 𝑌)), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ifcif 4488 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 Fincfn 8918 Basecbs 17179 +_gcplusg 17220 ._rcmulr 17221 0_gc0g 17402 Grpcgrp 18865 Ringcrg 20142 CRingccrg 20143 Mat cmat 22294 maDet cmdat 22471

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-ot 4598 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-word 14479 df-lsw 14528 df-concat 14536 df-s1 14561 df-substr 14606 df-pfx 14636 df-splice 14715 df-reverse 14724 df-s2 14814 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 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-efmnd 18796 df-grp 18868 df-minusg 18869 df-mulg 19000 df-subg 19055 df-ghm 19145 df-gim 19191 df-cntz 19249 df-oppg 19278 df-symg 19300 df-pmtr 19372 df-psgn 19421 df-evpm 19422 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-drng 20640 df-sra 21080 df-rgmod 21081 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-dsmm 21641 df-frlm 21656 df-mat 22295 df-mdet 22472

This theorem is referenced by: maducoeval2 22527 matunitlindflem1 37610

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