mdetpmtr2 - Mathbox for Thierry Arnoux

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Theorem mdetpmtr2 33770

Description: The determinant of a matrix with permuted columns is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)

**Hypotheses**
Ref	Expression
mdetpmtr.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetpmtr.b	⊢ 𝐵 = (Base‘𝐴)
mdetpmtr.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdetpmtr.g	⊢ 𝐺 = (Base‘(SymGrp‘𝑁))
mdetpmtr.s	⊢ 𝑆 = (pmSgn‘𝑁)
mdetpmtr.z	⊢ 𝑍 = (ℤRHom‘𝑅)
mdetpmtr.t	⊢ · = (._r‘𝑅)
mdetpmtr2.e	⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗)))

**Assertion**
Ref	Expression
mdetpmtr2	⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))

Distinct variable groups: 𝐵,𝑖,𝑗 𝑖,𝐺,𝑗 𝑖,𝑀,𝑗 𝑖,𝑁,𝑗 𝑃,𝑖,𝑗 𝑅,𝑖,𝑗 Allowed substitution hints: 𝐴(𝑖,𝑗) 𝐷(𝑖,𝑗) 𝑆(𝑖,𝑗) · (𝑖,𝑗) 𝐸(𝑖,𝑗) 𝑍(𝑖,𝑗)

**Proof of Theorem mdetpmtr2**
Step	Hyp	Ref	Expression
1		simpll 766	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ CRing)
2		simplr 768	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑁 ∈ Fin)
3		mdetpmtr.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4		mdetpmtr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
5	3, 4	mattposcl 22480	. . . 4 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
6	5	ad2antrl 727	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → tpos 𝑀 ∈ 𝐵)
7		simprr 772	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑃 ∈ 𝐺)
8		mdetpmtr.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
9		mdetpmtr.g	. . . 4 ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))
10		mdetpmtr.s	. . . 4 ⊢ 𝑆 = (pmSgn‘𝑁)
11		mdetpmtr.z	. . . 4 ⊢ 𝑍 = (ℤRHom‘𝑅)
12		mdetpmtr.t	. . . 4 ⊢ · = (._r‘𝑅)
13		mdetpmtr2.e	. . . . . 6 ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗)))
14		ovtpos 8282	. . . . . . . . 9 ⊢ ((𝑃‘𝑗)tpos 𝑀𝑖) = (𝑖𝑀(𝑃‘𝑗))
15	14	eqcomi 2749	. . . . . . . 8 ⊢ (𝑖𝑀(𝑃‘𝑗)) = ((𝑃‘𝑗)tpos 𝑀𝑖)
16	15	a1i 11	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀(𝑃‘𝑗)) = ((𝑃‘𝑗)tpos 𝑀𝑖))
17	16	mpoeq3ia 7528	. . . . . 6 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
18	13, 17	eqtri 2768	. . . . 5 ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
19	18	tposmpo 8304	. . . 4 ⊢ tpos 𝐸 = (𝑗 ∈ 𝑁, 𝑖 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
20	3, 4, 8, 9, 10, 11, 12, 19	mdetpmtr1 33769	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (tpos 𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘tpos 𝐸)))
21	1, 2, 6, 7, 20	syl22anc 838	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘tpos 𝐸)))
22	8, 3, 4	mdettpos 22638	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
23	22	ad2ant2r 746	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
24		eqid 2740	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
25		simp2 1137	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
26	7	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ 𝐺)
27		simp3 1138	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
28		eqid 2740	. . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
29	28, 9	symgfv 19421	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
30	26, 27, 29	syl2anc 583	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
31		simp1rl 1238	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
32	3, 24, 4, 25, 30, 31	matecld 22453	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀(𝑃‘𝑗)) ∈ (Base‘𝑅))
33	3, 24, 4, 2, 1, 32	matbas2d 22450	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) ∈ 𝐵)
34	13, 33	eqeltrid 2848	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐸 ∈ 𝐵)
35	8, 3, 4	mdettpos 22638	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐸 ∈ 𝐵) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
36	1, 34, 35	syl2anc 583	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
37	36	oveq2d 7464	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘tpos 𝐸)) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))
38	21, 23, 37	3eqtr3d 2788	1 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∘ ccom 5704 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 tpos ctpos 8266 Fincfn 9003 Basecbs 17258 ._rcmulr 17312 SymGrpcsymg 19410 pmSgncpsgn 19531 CRingccrg 20261 ℤRHomczrh 21533 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-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-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: mdetpmtr12 33771

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