mdetpmtr2 - Mathbox for Thierry Arnoux

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Theorem mdetpmtr2 34081

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 776	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → 𝑅 ∈ CRing)
2		simplr 778	. . 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 22500	. . . 4 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
6	5	ad2antrl 738	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → tpos 𝑀 ∈ 𝐵)
7		simprr 782	. . 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 8214	. . . . . . . . 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 7468	. . . . . 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 8236	. . . 4 ⊢ tpos 𝐸 = (𝑗 ∈ 𝑁, 𝑖 ∈ 𝑁 ↦ ((𝑃‘𝑗)tpos 𝑀𝑖))
20	3, 4, 8, 9, 10, 11, 12, 19	mdetpmtr1 34080	. . 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 849	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘tpos 𝐸)))
22	8, 3, 4	mdettpos 22658	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
23	22	ad2ant2r 757	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
24		eqid 2761	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
25		simp2 1149	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
26	7	3ad2ant1 1145	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ 𝐺)
27		simp3 1150	. . . . . . . 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 19410	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
30	26, 27, 29	syl2anc 593	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃‘𝑗) ∈ 𝑁)
31		simp1rl 1251	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ 𝐵)
32	3, 24, 4, 25, 30, 31	matecld 22473	. . . . . 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 22470	. . . . 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 22658	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐸 ∈ 𝐵) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
36	1, 34, 35	syl2anc 593	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘tpos 𝐸) = (𝐷‘𝐸))
37	36	oveq2d 7406	. 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

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∘ ccom 5647 ‘cfv 6515 (class class class)co 7390 ∈ cmpo 7392 tpos ctpos 8198 Fincfn 8920 Basecbs 17235 ._rcmulr 17277 SymGrpcsymg 19399 pmSgncpsgn 19519 CRingccrg 20270 ℤRHomczrh 21538 Mat cmat 22454 maDet cmdat 22631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-addf 11145 ax-mulf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-xor 1531 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-tpos 8199 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-xnn0 12548 df-z 12562 df-dec 12682 df-uz 12833 df-rp 12987 df-fz 13506 df-fzo 13653 df-seq 14008 df-exp 14068 df-hash 14337 df-word 14520 df-lsw 14569 df-concat 14577 df-s1 14603 df-substr 14648 df-pfx 14678 df-splice 14756 df-reverse 14765 df-s2 14854 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-efmnd 18893 df-grp 18968 df-minusg 18969 df-mulg 19100 df-subg 19155 df-ghm 19244 df-gim 19289 df-cntz 19347 df-oppg 19376 df-symg 19400 df-pmtr 19472 df-psgn 19521 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-cring 20272 df-oppr 20372 df-dvdsr 20392 df-unit 20393 df-invr 20423 df-dvr 20436 df-rhm 20507 df-subrng 20582 df-subrg 20606 df-drng 20767 df-sra 21227 df-rgmod 21228 df-cnfld 21412 df-zring 21486 df-zrh 21542 df-dsmm 21771 df-frlm 21786 df-mat 22455 df-mdet 22632

This theorem is referenced by: mdetpmtr12 34082

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