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Theorem mdetf 22533

Description: Functionality of the determinant, see also definition in [Lang] p. 513. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)

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

**Assertion**
Ref	Expression
mdetf	⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶𝐾)

Proof of Theorem mdetf Dummy variables 𝑝 𝑐 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetf.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
2		crngring 20205	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	adantr 480	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ Ring)
4		ringcmn 20242	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
5	3, 4	syl 17	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ CMnd)
6		mdetf.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
7		mdetf.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
8	6, 7	matrcl 22350	. . . . . 6 ⊢ (𝑚 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
9	8	adantl 481	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
10	9	simpld 494	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑁 ∈ Fin)
11		eqid 2735	. . . . 5 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
12		eqid 2735	. . . . 5 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
13	11, 12	symgbasfi 19360	. . . 4 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
14	10, 13	syl 17	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (Base‘(SymGrp‘𝑁)) ∈ Fin)
15	2	ad2antrr 726	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
16		zrhpsgnmhm 21544	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
17	3, 10, 16	syl2anc 584	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
18		eqid 2735	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
19	18, 1	mgpbas 20105	. . . . . . . 8 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
20	12, 19	mhmf 18767	. . . . . . 7 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
21	17, 20	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
22	21	ffvelcdmda 7074	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾)
23	18	crngmgp 20201	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
24	23	ad2antrr 726	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
25	10	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 ∈ Fin)
26	6, 1, 7	matbas2i 22360	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐵 → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
27	26	ad3antlr 731	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
28		elmapi 8863	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑚:(𝑁 × 𝑁)⟶𝐾)
29	27, 28	syl 17	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑚:(𝑁 × 𝑁)⟶𝐾)
30	11, 12	symgbasf 19357	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) → 𝑝:𝑁⟶𝑁)
31	30	adantl 481	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁⟶𝑁)
32	31	ffvelcdmda 7074	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → (𝑝‘𝑐) ∈ 𝑁)
33		simpr 484	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
34	29, 32, 33	fovcdmd 7579	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
35	34	ralrimiva 3132	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑐 ∈ 𝑁 ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
36	19, 24, 25, 35	gsummptcl 19948	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐))) ∈ 𝐾)
37		eqid 2735	. . . . . 6 ⊢ (._r‘𝑅) = (._r‘𝑅)
38	1, 37	ringcl 20210	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾 ∧ ((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐))) ∈ 𝐾) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐)))) ∈ 𝐾)
39	15, 22, 36, 38	syl3anc 1373	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐)))) ∈ 𝐾)
40	39	ralrimiva 3132	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → ∀𝑝 ∈ (Base‘(SymGrp‘𝑁))((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐)))) ∈ 𝐾)
41	1, 5, 14, 40	gsummptcl 19948	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐)))))) ∈ 𝐾)
42		mdetf.d	. . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅)
43		eqid 2735	. . 3 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
44		eqid 2735	. . 3 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
45	42, 6, 7, 12, 43, 44, 37, 18	mdetfval 22524	. 2 ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(._r‘𝑅)((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐)))))))
46	41, 45	fmptd 7104	1 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ↦ cmpt 5201 × cxp 5652 ∘ ccom 5658 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ↑_m cmap 8840 Fincfn 8959 Basecbs 17228 ._rcmulr 17272 Σ_g cgsu 17454 MndHom cmhm 18759 SymGrpcsymg 19350 pmSgncpsgn 19470 CMndccmn 19761 mulGrpcmgp 20100 Ringcrg 20193 CRingccrg 20194 ℤRHomczrh 21460 Mat cmat 22345 maDet cmdat 22522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-ot 4610 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-splice 14768 df-reverse 14777 df-s2 14867 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-0g 17455 df-gsum 17456 df-prds 17461 df-pws 17463 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-efmnd 18847 df-grp 18919 df-minusg 18920 df-mulg 19051 df-subg 19106 df-ghm 19196 df-gim 19242 df-cntz 19300 df-oppg 19329 df-symg 19351 df-pmtr 19423 df-psgn 19472 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-drng 20691 df-sra 21131 df-rgmod 21132 df-cnfld 21316 df-zring 21408 df-zrh 21464 df-dsmm 21692 df-frlm 21707 df-mat 22346 df-mdet 22523

This theorem is referenced by: mdetcl 22534 mdetr0 22543 mdetero 22548 mdetuni0 22559 mdetmul 22561 maduf 22579 madurid 22582 madulid 22583 matunit 22616 cramerimp 22624

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