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 22659

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 8215	. . . . . . 7 ⊢ ((𝑝‘𝑥)tpos 𝑀𝑥) = (𝑥𝑀(𝑝‘𝑥))
2	1	mpteq2i 5193	. . . . . 6 ⊢ (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥)) = (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥)))
3	2	oveq2i 7402	. . . . 5 ⊢ ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥))) = ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥))))
4	3	oveq2i 7402	. . . 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 5193	. . 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 7402	. 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 22501	. . . 4 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵)
10	9	adantl 485	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → tpos 𝑀 ∈ 𝐵)
11		mdettpos.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
12		eqid 2761	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
13		eqid 2761	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
14		eqid 2761	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
15		eqid 2761	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
16		eqid 2761	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17	11, 7, 8, 12, 13, 14, 15, 16	mdetleib 22635	. . 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 22636	. 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 2822	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ↦ cmpt 5178 ∘ ccom 5647 ‘cfv 6516 (class class class)co 7391 tpos ctpos 8199 Basecbs 17236 ._rcmulr 17278 Σ_g cgsu 17460 SymGrpcsymg 19400 pmSgncpsgn 19520 mulGrpcmgp 20177 CRingccrg 20271 ℤRHomczrh 21539 Mat cmat 22455 maDet cmdat 22632

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 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-addf 11146 ax-mulf 11147

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 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-tpos 8200 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-sup 9382 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-xnn0 12549 df-z 12563 df-dec 12683 df-uz 12834 df-rp 12988 df-fz 13507 df-fzo 13654 df-seq 14009 df-exp 14069 df-hash 14338 df-word 14521 df-lsw 14570 df-concat 14578 df-s1 14604 df-substr 14649 df-pfx 14679 df-splice 14757 df-reverse 14766 df-s2 14855 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-0g 17461 df-gsum 17462 df-prds 17467 df-pws 17469 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-efmnd 18894 df-grp 18969 df-minusg 18970 df-mulg 19101 df-subg 19156 df-ghm 19245 df-gim 19290 df-cntz 19348 df-oppg 19377 df-symg 19401 df-pmtr 19473 df-psgn 19522 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-cring 20273 df-oppr 20373 df-dvdsr 20393 df-unit 20394 df-invr 20424 df-dvr 20437 df-rhm 20508 df-subrng 20583 df-subrg 20607 df-drng 20768 df-sra 21228 df-rgmod 21229 df-cnfld 21413 df-zring 21487 df-zrh 21543 df-dsmm 21772 df-frlm 21787 df-mat 22456 df-mdet 22633

This theorem is referenced by: madutpos 22690 madulid 22693 mdetpmtr2 34082

W3C validator