mdetpmtr2 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mdetpmtr2		Structured version Visualization version GIF version

Theorem mdetpmtr2 33827

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 22361	. . . 4 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
6	5	ad2antrl 728	. . 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 8166	. . . . . . . . 9 ⊢ ((𝑃‘𝑗)tpos 𝑀𝑖) = (𝑖𝑀(𝑃‘𝑗))
15	14	eqcomi 2739	. . . . . . . 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 7419	. . . . . 6 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
18	13, 17	eqtri 2753	. . . . 5 ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
19	18	tposmpo 8188	. . . 4 ⊢ tpos 𝐸 = (𝑗 ∈ 𝑁, 𝑖 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
20	3, 4, 8, 9, 10, 11, 12, 19	mdetpmtr1 33826	. . 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 22519	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
23	22	ad2ant2r 747	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
24		eqid 2730	. . . . . 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 2730	. . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
29	28, 9	symgfv 19285	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
30	26, 27, 29	syl2anc 584	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
31		simp1rl 1239	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
32	3, 24, 4, 25, 30, 31	matecld 22334	. . . . . 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 22331	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) ∈ 𝐵)
34	13, 33	eqeltrid 2833	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐸 ∈ 𝐵)
35	8, 3, 4	mdettpos 22519	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐸 ∈ 𝐵) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
36	1, 34, 35	syl2anc 584	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
37	36	oveq2d 7357	. 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 2773	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 1086 = wceq 1541 ∈ wcel 2110 ∘ ccom 5618 ‘cfv 6477 (class class class)co 7341 ∈ cmpo 7343 tpos ctpos 8150 Fincfn 8864 Basecbs 17112 ._rcmulr 17154 SymGrpcsymg 19274 pmSgncpsgn 19394 CRingccrg 20145 ℤRHomczrh 21429 Mat cmat 22315 maDet cmdat 22492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-addf 11077 ax-mulf 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1513 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-ot 4583 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-sup 9321 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-xnn0 12447 df-z 12461 df-dec 12581 df-uz 12725 df-rp 12883 df-fz 13400 df-fzo 13547 df-seq 13901 df-exp 13961 df-hash 14230 df-word 14413 df-lsw 14462 df-concat 14470 df-s1 14496 df-substr 14541 df-pfx 14571 df-splice 14649 df-reverse 14658 df-s2 14747 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-0g 17337 df-gsum 17338 df-prds 17343 df-pws 17345 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-efmnd 18769 df-grp 18841 df-minusg 18842 df-mulg 18973 df-subg 19028 df-ghm 19118 df-gim 19164 df-cntz 19222 df-oppg 19251 df-symg 19275 df-pmtr 19347 df-psgn 19396 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-oppr 20248 df-dvdsr 20268 df-unit 20269 df-invr 20299 df-dvr 20312 df-rhm 20383 df-subrng 20454 df-subrg 20478 df-drng 20639 df-sra 21100 df-rgmod 21101 df-cnfld 21285 df-zring 21377 df-zrh 21433 df-dsmm 21662 df-frlm 21677 df-mat 22316 df-mdet 22493

This theorem is referenced by: mdetpmtr12 33828

W3C validator