mdetero - Metamath Proof Explorer

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Theorem mdetero 22905

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 20452	. . . . . 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 20457	. . . 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 4529	. . 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 22902	. 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 22899	. . . 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 2761	. . . . . 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 22904	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑌, if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
27	26	oveq2d 7428	. . . 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 20505	. . . . 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 2800	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑊 · 𝑌), if(𝑖 = 𝐽, 𝑌, 𝑍)))) = (0_g‘𝑅))
31	30	oveq2d 7428	. 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 20444	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
33	9, 32	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
34		eqid 2761	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
35		eqid 2761	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
36	1, 34, 35, 2	mdetf 22890	. . . . 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 4529	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)) ∈ 𝐾)
39	34, 2, 35, 5, 4, 38	matbas2d 22718	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍))) ∈ (Base‘(𝑁 Mat 𝑅)))
40	37, 39	ffvelcdmd 7077	. . 3 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑌, 𝑍)))) ∈ 𝐾)
41	2, 3, 23	grprid 19159	. . 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 2800	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 2956 ifcif 4482 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 Fincfn 8957 Basecbs 17367 +_gcplusg 17408 ._rcmulr 17409 0_gc0g 17590 Grpcgrp 19124 Ringcrg 20439 CRingccrg 20440 Mat cmat 22702 maDet cmdat 22879

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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-addf 11260 ax-mulf 11261

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-sup 9418 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-xnn0 12661 df-z 12675 df-dec 12796 df-uz 12947 df-rp 13102 df-fz 13621 df-fzo 13769 df-seq 14125 df-exp 14185 df-hash 14455 df-word 14639 df-lsw 14688 df-concat 14696 df-s1 14723 df-substr 14769 df-pfx 14801 df-splice 14879 df-reverse 14888 df-s2 14979 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-0g 17592 df-gsum 17593 df-prds 17598 df-pws 17600 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-mhm 18958 df-submnd 18959 df-efmnd 19045 df-grp 19127 df-minusg 19128 df-mulg 19258 df-subg 19313 df-ghm 19408 df-gim 19453 df-cntz 19511 df-oppg 19540 df-symg 19564 df-pmtr 19636 df-psgn 19685 df-evpm 19686 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-cring 20442 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-rhm 20682 df-subrng 20778 df-subrg 20802 df-drng 20962 df-sra 21428 df-rgmod 21429 df-cnfld 21659 df-zring 21733 df-zrh 21789 df-dsmm 22018 df-frlm 22033 df-mat 22703 df-mdet 22880

This theorem is used by: maducoeval2 22935 matunitlindflem1 22974

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