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Theorem mdetf 22721

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 20327	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	adantr 485	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ Ring)
4		ringcmn 20365	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
5	3, 4	syl 18	. . 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 22538	. . . . . 6 ⊢ (𝑚 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
9	8	adantl 486	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
10	9	simpld 499	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑁 ∈ Fin)
11		eqid 2769	. . . . 5 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
12		eqid 2769	. . . . 5 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
13	11, 12	symgbasfi 19449	. . . 4 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
14	10, 13	syl 18	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (Base‘(SymGrp‘𝑁)) ∈ Fin)
15	2	ad2antrr 738	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
16		zrhpsgnmhm 21703	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
17	3, 10, 16	syl2anc 595	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
18		eqid 2769	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
19	18, 1	mgpbas 20221	. . . . . . . 8 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
20	12, 19	mhmf 18847	. . . . . . 7 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
21	17, 20	syl 18	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
22	21	ffvelcdmda 7080	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾)
23	18	crngmgp 20323	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
24	23	ad2antrr 738	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
25	10	adantr 485	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 ∈ Fin)
26	6, 1, 7	matbas2i 22548	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐵 → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
27	26	ad3antlr 743	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
28		elmapi 8846	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑚:(𝑁 × 𝑁)⟶𝐾)
29	27, 28	syl 18	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑚:(𝑁 × 𝑁)⟶𝐾)
30	11, 12	symgbasf 19446	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) → 𝑝:𝑁⟶𝑁)
31	30	adantl 486	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁⟶𝑁)
32	31	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → (𝑝‘𝑐) ∈ 𝑁)
33		simpr 489	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
34	29, 32, 33	fovcdmd 7583	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
35	34	ralrimiva 3163	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑐 ∈ 𝑁 ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
36	19, 24, 25, 35	gsummptcl 20037	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐))) ∈ 𝐾)
37		eqid 2769	. . . . . 6 ⊢ (._r‘𝑅) = (._r‘𝑅)
38	1, 37	ringcl 20332	. . . . 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 1396	. . . 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 3163	. . 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 20037	. 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 2769	. . 3 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
44		eqid 2769	. . 3 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
45	42, 6, 7, 12, 43, 44, 37, 18	mdetfval 22712	. 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 7110	1 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ↦ cmpt 5196 × cxp 5660 ∘ ccom 5666 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ↑_m cmap 8824 Fincfn 8943 Basecbs 17269 ._rcmulr 17311 Σ_g cgsu 17493 MndHom cmhm 18839 SymGrpcsymg 19439 pmSgncpsgn 19559 CMndccmn 19850 mulGrpcmgp 20216 Ringcrg 20315 CRingccrg 20316 ℤRHomczrh 21618 Mat cmat 22533 maDet cmdat 22710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-addf 11179 ax-mulf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-xor 1539 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-ot 4603 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-tpos 8222 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-sup 9402 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 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 12505 df-xnn0 12578 df-z 12592 df-dec 12712 df-uz 12863 df-rp 13017 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-word 14551 df-lsw 14600 df-concat 14608 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14787 df-reverse 14796 df-s2 14885 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-0g 17494 df-gsum 17495 df-prds 17500 df-pws 17502 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-mhm 18841 df-submnd 18842 df-efmnd 18928 df-grp 19003 df-minusg 19004 df-mulg 19134 df-subg 19189 df-ghm 19284 df-gim 19329 df-cntz 19387 df-oppg 19416 df-symg 19440 df-pmtr 19512 df-psgn 19561 df-cmn 19852 df-abl 19853 df-mgp 20217 df-rng 20231 df-ur 20264 df-ring 20317 df-cring 20318 df-oppr 20419 df-dvdsr 20439 df-unit 20440 df-invr 20470 df-dvr 20483 df-rhm 20554 df-subrng 20631 df-subrg 20655 df-drng 20815 df-sra 21272 df-rgmod 21273 df-cnfld 21492 df-zring 21566 df-zrh 21622 df-dsmm 21851 df-frlm 21866 df-mat 22534 df-mdet 22711

This theorem is referenced by: mdetcl 22722 mdetr0 22731 mdetero 22736 mdetuni0 22747 mdetmul 22749 maduf 22767 madurid 22770 madulid 22771 matunit 22804 cramerimp 22812

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