mdetero - Metamath Proof Explorer

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Theorem mdetero 22838

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 1149	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
8		crngring 20390	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	4, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	9	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
11		mdetero.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐾)
12	11	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑊 ∈ 𝐾)
13		mdetero.y	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
14	13	3adant2 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
15		mdetero.t	. . . . 5 ⊢ · = (._r‘𝑅)
16	2, 15	ringcl 20395	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑊 · 𝑌) ∈ 𝐾)
17	10, 12, 14, 16	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑊 · 𝑌) ∈ 𝐾)
18		mdetero.z	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ 𝐾)
19	14, 18	ifcld 4532	. . 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 22835	. 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 22832	. . . 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 2762	. . . . . 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 22837	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
27	26	oveq2d 7433	. . . 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 20442	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐾) → (𝑊 · (0_g‘𝑅)) = (0_g‘𝑅))
29	9, 11, 28	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑊 · (0_g‘𝑅)) = (0_g‘𝑅))
30	22, 27, 29	3eqtrd 2801	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
31	30	oveq2d 7433	. 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 20383	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
33	9, 32	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
34		eqid 2762	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
35		eqid 2762	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
36	1, 34, 35, 2	mdetf 22823	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾)
37	4, 36	syl 18	. . . 4 ⊢ (𝜑 → 𝐷:(Base‘(𝑁 Mat 𝑅))⟶𝐾)
38	7, 19	ifcld 4532	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)) ∈ 𝐾)
39	34, 2, 35, 5, 4, 38	matbas2d 22651	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))) ∈ (Base‘(𝑁 Mat 𝑅)))
40	37, 39	ffvelcdmd 7082	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) ∈ 𝐾)
41	2, 3, 23	grprid 19098	. . 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 596	. 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 2801	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ifcif 4485 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 ∈ cmpo 7419 Fincfn 8956 Basecbs 17307 +_gcplusg 17348 ._rcmulr 17349 0_gc0g 17530 Grpcgrp 19063 Ringcrg 20378 CRingccrg 20379 Mat cmat 22635 maDet cmdat 22812

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-addf 11207 ax-mulf 11208

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-sup 9416 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-xnn0 12606 df-z 12620 df-dec 12741 df-uz 12892 df-rp 13047 df-fz 13566 df-fzo 13714 df-seq 14070 df-exp 14130 df-hash 14399 df-word 14583 df-lsw 14632 df-concat 14640 df-s1 14667 df-substr 14713 df-pfx 14745 df-splice 14823 df-reverse 14832 df-s2 14923 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-0g 17532 df-gsum 17533 df-prds 17538 df-pws 17540 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-submnd 18898 df-efmnd 18984 df-grp 19066 df-minusg 19067 df-mulg 19197 df-subg 19252 df-ghm 19347 df-gim 19392 df-cntz 19450 df-oppg 19479 df-symg 19503 df-pmtr 19575 df-psgn 19624 df-evpm 19625 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-cring 20381 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-rhm 20619 df-subrng 20714 df-subrg 20738 df-drng 20898 df-sra 21363 df-rgmod 21364 df-cnfld 21592 df-zring 21666 df-zrh 21722 df-dsmm 21951 df-frlm 21966 df-mat 22636 df-mdet 22813

This theorem is used by: maducoeval2 22868 matunitlindflem1 22907

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