mdetralt2 - Metamath Proof Explorer

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Theorem mdetralt2 22631

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 2735	. 2 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2735	. 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 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾)
10		mdetralt2.y	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾)
11	9, 10	ifcld 4577	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐽, 𝑋, 𝑌) ∈ 𝐾)
12	9, 11	ifcld 4577	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) ∈ 𝐾)
13	2, 6, 3, 7, 5, 12	matbas2d 22445	. 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 2736	. . . . 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 4537	. . . . . . 7 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
19	18	ad2antrl 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
20		csbeq1a 3922	. . . . . . 7 ⊢ (𝑗 = 𝑤 → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
21	20	ad2antll 729	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
22	19, 21	eqtrd 2775	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
23		eqidd 2736	. . . . 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 1912	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑤 ∈ 𝑁)
27		nfcsb1v 3933	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋
28	27	nfel1 2920	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾
29	26, 28	nfim 1894	. . . . . 6 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)
30		eleq1w 2822	. . . . . . . 8 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑁 ↔ 𝑤 ∈ 𝑁))
31	30	anbi2d 630	. . . . . . 7 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑁) ↔ (𝜑 ∧ 𝑤 ∈ 𝑁)))
32	20	eleq1d 2824	. . . . . . 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 2238	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)
35		nfv 1912	. . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑤 ∈ 𝑁)
36		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑗𝐼
37		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑖𝑤
38		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑗⦌𝑋
39	17, 22, 23, 24, 25, 34, 35, 26, 36, 37, 38, 27	ovmpodxf 7583	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = ⦋𝑤 / 𝑗⦌𝑋)
40		iftrue 4537	. . . . . . . . 9 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐽, 𝑋, 𝑌) = 𝑋)
41	40	ifeq2d 4551	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = if(𝑖 = 𝐼, 𝑋, 𝑋))
42		ifid 4571	. . . . . . . 8 ⊢ if(𝑖 = 𝐼, 𝑋, 𝑋) = 𝑋
43	41, 42	eqtrdi 2791	. . . . . . 7 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
44	43	ad2antrl 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋)
45	20	ad2antll 729	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋)
46	44, 45	eqtrd 2775	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
47		eqidd 2736	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐽) → 𝑁 = 𝑁)
48	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐽 ∈ 𝑁)
49		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑗𝐽
50	17, 46, 47, 48, 25, 34, 35, 26, 49, 37, 38, 27	ovmpodxf 7583	. . . 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 2778	. . 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 3144	. 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 22630	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 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⦋csb 3908 ifcif 4531 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8984 Basecbs 17245 0_gc0g 17486 CRingccrg 20252 Mat cmat 22427 maDet cmdat 22606

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-efmnd 18895 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-ghm 19244 df-gim 19290 df-cntz 19348 df-oppg 19377 df-symg 19402 df-pmtr 19475 df-psgn 19524 df-evpm 19525 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-dsmm 21770 df-frlm 21785 df-mat 22428 df-mdet 22607

This theorem is referenced by: mdetero 22632 madurid 22666

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