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Theorem mdetralt2 20463

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 2651	. 2 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2651	. 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 1100	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
10		mdetralt2.y	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
11	9, 10	ifcld 4164	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐽, 𝑋, 𝑌) ∈ 𝐾)
12	9, 11	ifcld 4164	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) ∈ 𝐾)
13	2, 6, 3, 7, 5, 12	matbas2d 20277	. 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 2652	. . . . 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 4125	. . . . . . 7 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
19	18	ad2antrl 764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
20		csbeq1a 3575	. . . . . . 7 ⊢ (𝑗 = 𝑤 → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
21	20	ad2antll 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
22	19, 21	eqtrd 2685	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
23		eqidd 2652	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐼) → 𝑁 = 𝑁)
24	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐼 ∈ 𝑁)
25		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝑤 ∈ 𝑁)
26		nfv 1883	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑤 ∈ 𝑁)
27		nfcsb1v 3582	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋
28	27	nfel1 2808	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾
29	26, 28	nfim 1865	. . . . . 6 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)
30		eleq1 2718	. . . . . . . 8 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑁 ↔ 𝑤 ∈ 𝑁))
31	30	anbi2d 740	. . . . . . 7 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑁) ↔ (𝜑 ∧ 𝑤 ∈ 𝑁)))
32	20	eleq1d 2715	. . . . . . 7 ⊢ (𝑗 = 𝑤 → (𝑋 ∈ 𝐾 ↔ ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾))
33	31, 32	imbi12d 333	. . . . . 6 ⊢ (𝑗 = 𝑤 → (((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)))
34	29, 33, 8	chvar 2298	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)
35		nfv 1883	. . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑤 ∈ 𝑁)
36		nfcv 2793	. . . . 5 ⊢ Ⅎ𝑗𝐼
37		nfcv 2793	. . . . 5 ⊢ Ⅎ𝑖𝑤
38		nfcv 2793	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑗⦌𝑋
39	17, 22, 23, 24, 25, 34, 35, 26, 36, 37, 38, 27	ovmpt2dxf 6828	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = ⦋𝑤 / 𝑗⦌𝑋)
40		iftrue 4125	. . . . . . . . 9 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐽, 𝑋, 𝑌) = 𝑋)
41	40	ifeq2d 4138	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = if(𝑖 = 𝐼, 𝑋, 𝑋))
42		ifid 4158	. . . . . . . 8 ⊢ if(𝑖 = 𝐼, 𝑋, 𝑋) = 𝑋
43	41, 42	syl6eq 2701	. . . . . . 7 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
44	43	ad2antrl 764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
45	20	ad2antll 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
46	44, 45	eqtrd 2685	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
47		eqidd 2652	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐽) → 𝑁 = 𝑁)
48	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐽 ∈ 𝑁)
49		nfcv 2793	. . . . 5 ⊢ Ⅎ𝑗𝐽
50	17, 46, 47, 48, 25, 34, 35, 26, 49, 37, 38, 27	ovmpt2dxf 6828	. . . 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 2688	. . 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 2995	. 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 20462	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 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ⦋csb 3566 ifcif 4119 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 Fincfn 7997 Basecbs 15904 0_gc0g 16147 CRingccrg 18594 Mat cmat 20261 maDet cmdat 20438

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-xor 1505 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-sup 8389 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-xnn0 11402 df-z 11416 df-dec 11532 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-word 13331 df-lsw 13332 df-concat 13333 df-s1 13334 df-substr 13335 df-splice 13336 df-reverse 13337 df-s2 13639 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-0g 16149 df-gsum 16150 df-prds 16155 df-pws 16157 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-mulg 17588 df-subg 17638 df-ghm 17705 df-gim 17748 df-cntz 17796 df-oppg 17822 df-symg 17844 df-pmtr 17908 df-psgn 17957 df-evpm 17958 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-rnghom 18763 df-drng 18797 df-subrg 18826 df-sra 19220 df-rgmod 19221 df-cnfld 19795 df-zring 19867 df-zrh 19900 df-dsmm 20124 df-frlm 20139 df-mat 20262 df-mdet 20439

This theorem is referenced by: mdetero 20464 madurid 20498

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