mdetralt2 - Metamath Proof Explorer

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Theorem mdetralt2 22494

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 2729	. 2 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3		eqid 2729	. 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 4523	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐽, 𝑋, 𝑌) ∈ 𝐾)
12	9, 11	ifcld 4523	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) ∈ 𝐾)
13	2, 6, 3, 7, 5, 12	matbas2d 22308	. 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 2730	. . . . 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 4482	. . . . . . 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 3865	. . . . . . 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 2764	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
23		eqidd 2730	. . . . 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 1914	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑤 ∈ 𝑁)
27		nfcsb1v 3875	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋
28	27	nfel1 2908	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾
29	26, 28	nfim 1896	. . . . . 6 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)
30		eleq1w 2811	. . . . . . . 8 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑁 ↔ 𝑤 ∈ 𝑁))
31	30	anbi2d 630	. . . . . . 7 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑁) ↔ (𝜑 ∧ 𝑤 ∈ 𝑁)))
32	20	eleq1d 2813	. . . . . . 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 1914	. . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑤 ∈ 𝑁)
36		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑗𝐼
37		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑖𝑤
38		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑗⦌𝑋
39	17, 22, 23, 24, 25, 34, 35, 26, 36, 37, 38, 27	ovmpodxf 7499	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = ⦋𝑤 / 𝑗⦌𝑋)
40		iftrue 4482	. . . . . . . . 9 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐽, 𝑋, 𝑌) = 𝑋)
41	40	ifeq2d 4497	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = if(𝑖 = 𝐼, 𝑋, 𝑋))
42		ifid 4517	. . . . . . . 8 ⊢ if(𝑖 = 𝐼, 𝑋, 𝑋) = 𝑋
43	41, 42	eqtrdi 2780	. . . . . . 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 2764	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋)
47		eqidd 2730	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐽) → 𝑁 = 𝑁)
48	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐽 ∈ 𝑁)
49		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑗𝐽
50	17, 46, 47, 48, 25, 34, 35, 26, 49, 37, 38, 27	ovmpodxf 7499	. . . 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 2767	. . 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 3121	. 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 22493	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 1540 ∈ wcel 2109 ≠ wne 2925 ⦋csb 3851 ifcif 4476 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 Fincfn 8872 Basecbs 17120 0_gc0g 17343 CRingccrg 20119 Mat cmat 22292 maDet cmdat 22469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-addf 11088 ax-mulf 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-sup 9332 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14503 df-substr 14548 df-pfx 14578 df-splice 14656 df-reverse 14665 df-s2 14755 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-submnd 18658 df-efmnd 18743 df-grp 18815 df-minusg 18816 df-mulg 18947 df-subg 19002 df-ghm 19092 df-gim 19138 df-cntz 19196 df-oppg 19225 df-symg 19249 df-pmtr 19321 df-psgn 19370 df-evpm 19371 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-drng 20616 df-sra 21077 df-rgmod 21078 df-cnfld 21262 df-zring 21354 df-zrh 21410 df-dsmm 21639 df-frlm 21654 df-mat 22293 df-mdet 22470

This theorem is referenced by: mdetero 22495 madurid 22529

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