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Theorem mdettpos 22594

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 8181	. . . . . . 7 ⊢ ((𝑝‘𝑥)tpos 𝑀𝑥) = (𝑥𝑀(𝑝‘𝑥))
2	1	mpteq2i 5168	. . . . . 6 ⊢ (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥)) = (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥)))
3	2	oveq2i 7367	. . . . 5 ⊢ ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)tpos 𝑀𝑥))) = ((mulGrp‘𝑅) Σ_g (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥))))
4	3	oveq2i 7367	. . . 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 5168	. . 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 7367	. 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 22436	. . . 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 2739	. . . 4 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
13		eqid 2739	. . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
14		eqid 2739	. . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
15		eqid 2739	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
16		eqid 2739	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17	11, 7, 8, 12, 13, 14, 15, 16	mdetleib 22570	. . 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 22571	. 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 2800	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ↦ cmpt 5153 ∘ ccom 5622 ‘cfv 6485 (class class class)co 7356 tpos ctpos 8165 Basecbs 17170 ._rcmulr 17212 Σ_g cgsu 17394 SymGrpcsymg 19335 pmSgncpsgn 19455 mulGrpcmgp 20112 CRingccrg 20206 ℤRHomczrh 21474 Mat cmat 22390 maDet cmdat 22567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-xor 1519 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-ot 4564 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-word 14467 df-lsw 14516 df-concat 14524 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14703 df-reverse 14712 df-s2 14801 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-efmnd 18828 df-grp 18903 df-minusg 18904 df-mulg 19035 df-subg 19090 df-ghm 19179 df-gim 19225 df-cntz 19283 df-oppg 19312 df-symg 19336 df-pmtr 19408 df-psgn 19457 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-drng 20703 df-sra 21163 df-rgmod 21164 df-cnfld 21348 df-zring 21422 df-zrh 21478 df-dsmm 21707 df-frlm 21722 df-mat 22391 df-mdet 22568

This theorem is referenced by: madutpos 22625 madulid 22628 mdetpmtr2 34008

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