mdettpos - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mdettpos		Structured version Visualization version GIF version

Theorem mdettpos 21760

Description: Determinant is invariant under transposition. Proposition 4.8 in [Lang] p. 514. (Contributed by Stefan O'Rear, 9-Jul-2018.)

**Hypotheses**
Ref	Expression
mdettpos.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
mdettpos.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdettpos.b	⊢ 𝐵 = (Base‘𝐴)

**Assertion**
Ref	Expression
mdettpos	⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))

Proof of Theorem mdettpos Dummy variables 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovtpos 8057	. . . . . . 7 ⊢ ((𝑝‘𝑥)tpos 𝑀𝑥) = (𝑥𝑀(𝑝‘𝑥))
2	1	mpteq2i 5179	. . . . . 6 ⊢ (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥)) = (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥)))
3	2	oveq2i 7286	. . . . 5 ⊢ ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥))) = ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥))))
4	3	oveq2i 7286	. . . 4 ⊢ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥)))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥)))))
5	4	mpteq2i 5179	. . 3 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥))))) = (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥))))))
6	5	oveq2i 7286	. 2 ⊢ (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥)))))) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥)))))))
7		mdettpos.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
8		mdettpos.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
9	7, 8	mattposcl 21602	. . . 4 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
10	9	adantl 482	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos 𝑀 ∈ 𝐵)
11		mdettpos.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
12		eqid 2738	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
13		eqid 2738	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
14		eqid 2738	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
15		eqid 2738	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
16		eqid 2738	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17	11, 7, 8, 12, 13, 14, 15, 16	mdetleib 21736	. . 3 ⊢ (tpos 𝑀 ∈ 𝐵 → (𝐷‘tpos 𝑀) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥)))))))
18	10, 17	syl 17	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥)))))))
19	11, 7, 8, 12, 13, 14, 15, 16	mdetleib2 21737	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥))))))))
20	6, 18, 19	3eqtr4a 2804	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ↦ cmpt 5157 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 tpos ctpos 8041 Basecbs 16912 ._rcmulr 16963 Σ_g cgsu 17151 SymGrpcsymg 18974 pmSgncpsgn 19097 mulGrpcmgp 19720 CRingccrg 19784 ℤRHomczrh 20701 Mat cmat 21554 maDet cmdat 21733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-reverse 14472 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-efmnd 18508 df-grp 18580 df-minusg 18581 df-mulg 18701 df-subg 18752 df-ghm 18832 df-gim 18875 df-cntz 18923 df-oppg 18950 df-symg 18975 df-pmtr 19050 df-psgn 19099 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-dsmm 20939 df-frlm 20954 df-mat 21555 df-mdet 21734

This theorem is referenced by: madutpos 21791 madulid 21794 mdetpmtr2 31774

W3C validator