mdetralt2 - Metamath Proof Explorer

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Theorem mdetralt2 22636

Description: The determinant function is alternating regarding rows (matrix is given explicitly by its entries). (Contributed by SO, 16-Jul-2018.)

**Hypotheses**
Ref	Expression
mdetralt2.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetralt2.k	⊢ 𝐾 = (Base‘𝑅)
mdetralt2.z	⊢ 0 = (0_g‘𝑅)
mdetralt2.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mdetralt2.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetralt2.x	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
mdetralt2.y	⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
mdetralt2.i	⊢ (𝜑 → 𝐼 ∈ 𝑁)
mdetralt2.j	⊢ (𝜑 → 𝐽 ∈ 𝑁)
mdetralt2.ij	⊢ (𝜑 → 𝐼 ≠ 𝐽)

**Assertion**
Ref	Expression
mdetralt2	⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) = 0 )

Distinct variable groups: 𝜑,𝑖,𝑗 𝑖,𝐾,𝑗 𝑖,𝑁,𝑗 𝑖,𝐼,𝑗 𝑖,𝐽,𝑗 𝑖,𝑋 Allowed substitution hints: 𝐷(𝑖,𝑗) 𝑅(𝑖,𝑗) 𝑋(𝑗) 𝑌(𝑖,𝑗) 0 (𝑖,𝑗)

Proof of Theorem mdetralt2 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetralt2.d	. 2 ⊢ 𝐷 = (𝑁 maDet 𝑅)
2		eqid 2740	. 2 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2740	. 2 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
4		mdetralt2.z	. 2 ⊢ 0 = (0_g‘𝑅)
5		mdetralt2.r	. 2 ⊢ (𝜑 → 𝑅 ∈ CRing)
6		mdetralt2.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
7		mdetralt2.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
8		mdetralt2.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
9	8	3adant2 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
10		mdetralt2.y	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
11	9, 10	ifcld 4594	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐽, 𝑋, 𝑌) ∈ 𝐾)
12	9, 11	ifcld 4594	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) ∈ 𝐾)
13	2, 6, 3, 7, 5, 12	matbas2d 22450	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌))) ∈ (Base‘(𝑁 Mat 𝑅)))
14		mdetralt2.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑁)
15		mdetralt2.j	. 2 ⊢ (𝜑 → 𝐽 ∈ 𝑁)
16		mdetralt2.ij	. 2 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
17		eqidd 2741	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌))))
18		iftrue 4554	. . . . . . 7 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
19	18	ad2antrl 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
20		csbeq1a 3935	. . . . . . 7 ⊢ (𝑗 = 𝑤 → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
21	20	ad2antll 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
22	19, 21	eqtrd 2780	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
23		eqidd 2741	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐼) → 𝑁 = 𝑁)
24	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐼 ∈ 𝑁)
25		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝑤 ∈ 𝑁)
26		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑤 ∈ 𝑁)
27		nfcsb1v 3946	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋
28	27	nfel1 2925	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾
29	26, 28	nfim 1895	. . . . . 6 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)
30		eleq1w 2827	. . . . . . . 8 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑁 ↔ 𝑤 ∈ 𝑁))
31	30	anbi2d 629	. . . . . . 7 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑁) ↔ (𝜑 ∧ 𝑤 ∈ 𝑁)))
32	20	eleq1d 2829	. . . . . . 7 ⊢ (𝑗 = 𝑤 → (𝑋 ∈ 𝐾 ↔ ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾))
33	31, 32	imbi12d 344	. . . . . 6 ⊢ (𝑗 = 𝑤 → (((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)))
34	29, 33, 8	chvarfv 2241	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)
35		nfv 1913	. . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑤 ∈ 𝑁)
36		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑗𝐼
37		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑖𝑤
38		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑗⦌𝑋
39	17, 22, 23, 24, 25, 34, 35, 26, 36, 37, 38, 27	ovmpodxf 7600	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = ⦋𝑤 / 𝑗⦌𝑋)
40		iftrue 4554	. . . . . . . . 9 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐽, 𝑋, 𝑌) = 𝑋)
41	40	ifeq2d 4568	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = if(𝑖 = 𝐼, 𝑋, 𝑋))
42		ifid 4588	. . . . . . . 8 ⊢ if(𝑖 = 𝐼, 𝑋, 𝑋) = 𝑋
43	41, 42	eqtrdi 2796	. . . . . . 7 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
44	43	ad2antrl 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
45	20	ad2antll 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
46	44, 45	eqtrd 2780	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
47		eqidd 2741	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐽) → 𝑁 = 𝑁)
48	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐽 ∈ 𝑁)
49		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑗𝐽
50	17, 46, 47, 48, 25, 34, 35, 26, 49, 37, 38, 27	ovmpodxf 7600	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐽(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = ⦋𝑤 / 𝑗⦌𝑋)
51	39, 50	eqtr4d 2783	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = (𝐽(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤))
52	51	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑁 (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = (𝐽(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤))
53	1, 2, 3, 4, 5, 13, 14, 15, 16, 52	mdetralt 22635	1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ⦋csb 3921 ifcif 4548 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 Fincfn 9003 Basecbs 17258 0_gc0g 17499 CRingccrg 20261 Mat cmat 22432 maDet cmdat 22611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-word 14563 df-lsw 14611 df-concat 14619 df-s1 14644 df-substr 14689 df-pfx 14719 df-splice 14798 df-reverse 14807 df-s2 14897 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-efmnd 18904 df-grp 18976 df-minusg 18977 df-mulg 19108 df-subg 19163 df-ghm 19253 df-gim 19299 df-cntz 19357 df-oppg 19386 df-symg 19411 df-pmtr 19484 df-psgn 19533 df-evpm 19534 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-sra 21195 df-rgmod 21196 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-dsmm 21775 df-frlm 21790 df-mat 22433 df-mdet 22612

This theorem is referenced by: mdetero 22637 madurid 22671

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