mdetpmtr2 - Mathbox for Thierry Arnoux

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Theorem mdetpmtr2 34449

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 779	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ CRing)
2		simplr 781	. . 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 22761	. . . 4 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
6	5	ad2antrl 741	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → tpos 𝑀 ∈ 𝐵)
7		simprr 785	. . 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 8251	. . . . . . . . 9 ⊢ ((𝑃‘𝑗)tpos 𝑀𝑖) = (𝑖𝑀(𝑃‘𝑗))
15	14	eqcomi 2770	. . . . . . . 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 7496	. . . . . 6 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
18	13, 17	eqtri 2784	. . . . 5 ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
19	18	tposmpo 8273	. . . 4 ⊢ tpos 𝐸 = (𝑗 ∈ 𝑁, 𝑖 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
20	3, 4, 8, 9, 10, 11, 12, 19	mdetpmtr1 34448	. . 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 852	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘tpos 𝐸)))
22	8, 3, 4	mdettpos 22919	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
23	22	ad2ant2r 760	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
24		eqid 2761	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
25		simp2 1155	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
26	7	3ad2ant1 1151	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ 𝐺)
27		simp3 1156	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
28		eqid 2761	. . . . . . . . 9 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
29	28, 9	symgfv 19587	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
30	26, 27, 29	syl2anc 596	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
31		simp1rl 1257	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
32	3, 24, 4, 25, 30, 31	matecld 22734	. . . . . 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 22731	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) ∈ 𝐵)
34	13, 33	eqeltrid 2865	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝐸 ∈ 𝐵)
35	8, 3, 4	mdettpos 22919	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐸 ∈ 𝐵) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
36	1, 34, 35	syl2anc 596	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
37	36	oveq2d 7434	. 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 2804	1 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∘ ccom 5655 ‘cfv 6537 (class class class)co 7418 ∈ cmpo 7420 tpos ctpos 8235 Fincfn 8966 Basecbs 17380 ._rcmulr 17422 SymGrpcsymg 19576 pmSgncpsgn 19696 CRingccrg 20453 ℤRHomczrh 21798 Mat cmat 22715 maDet cmdat 22892

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-addf 11272 ax-mulf 11273

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-tpos 8236 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-sup 9427 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-xnn0 12673 df-z 12687 df-dec 12808 df-uz 12959 df-rp 13114 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-hash 14468 df-word 14652 df-lsw 14701 df-concat 14709 df-s1 14736 df-substr 14782 df-pfx 14814 df-splice 14892 df-reverse 14901 df-s2 14992 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-0g 17605 df-gsum 17606 df-prds 17611 df-pws 17613 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-mhm 18971 df-submnd 18972 df-efmnd 19058 df-grp 19140 df-minusg 19141 df-mulg 19271 df-subg 19326 df-ghm 19421 df-gim 19466 df-cntz 19524 df-oppg 19553 df-symg 19577 df-pmtr 19649 df-psgn 19698 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-cring 20455 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-dvr 20624 df-rhm 20695 df-subrng 20791 df-subrg 20815 df-drng 20975 df-sra 21441 df-rgmod 21442 df-cnfld 21672 df-zring 21746 df-zrh 21802 df-dsmm 22031 df-frlm 22046 df-mat 22716 df-mdet 22893

This theorem is used by: mdetpmtr12 34450

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