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Theorem mdetf 22517

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 20192	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	adantr 479	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ Ring)
4		ringcmn 20225	. . . 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 22332	. . . . . 6 ⊢ (𝑚 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
9	8	adantl 480	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
10	9	simpld 493	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑁 ∈ Fin)
11		eqid 2728	. . . . 5 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
12		eqid 2728	. . . . 5 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
13	11, 12	symgbasfi 19340	. . . 4 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
14	10, 13	syl 17	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (Base‘(SymGrp‘𝑁)) ∈ Fin)
15	2	ad2antrr 724	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
16		zrhpsgnmhm 21523	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
17	3, 10, 16	syl2anc 582	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
18		eqid 2728	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
19	18, 1	mgpbas 20087	. . . . . . . 8 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
20	12, 19	mhmf 18753	. . . . . . 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 7099	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾)
23	18	crngmgp 20188	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
24	23	ad2antrr 724	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
25	10	adantr 479	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 ∈ Fin)
26	6, 1, 7	matbas2i 22344	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐵 → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
27	26	ad3antlr 729	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
28		elmapi 8874	. . . . . . . . 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 19337	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) → 𝑝:𝑁⟶𝑁)
31	30	adantl 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁⟶𝑁)
32	31	ffvelcdmda 7099	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → (𝑝‘𝑐) ∈ 𝑁)
33		simpr 483	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
34	29, 32, 33	fovcdmd 7599	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
35	34	ralrimiva 3143	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑐 ∈ 𝑁 ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
36	19, 24, 25, 35	gsummptcl 19929	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐))) ∈ 𝐾)
37		eqid 2728	. . . . . 6 ⊢ (._r‘𝑅) = (._r‘𝑅)
38	1, 37	ringcl 20197	. . . . 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 1368	. . . 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 3143	. . 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 19929	. 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 2728	. . 3 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
44		eqid 2728	. . 3 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
45	42, 6, 7, 12, 43, 44, 37, 18	mdetfval 22508	. 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 7129	1 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ↦ cmpt 5235 × cxp 5680 ∘ ccom 5686 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ↑_m cmap 8851 Fincfn 8970 Basecbs 17187 ._rcmulr 17241 Σ_g cgsu 17429 MndHom cmhm 18745 SymGrpcsymg 19328 pmSgncpsgn 19451 CMndccmn 19742 mulGrpcmgp 20081 Ringcrg 20180 CRingccrg 20181 ℤRHomczrh 21432 Mat cmat 22327 maDet cmdat 22506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-xnn0 12583 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-word 14505 df-lsw 14553 df-concat 14561 df-s1 14586 df-substr 14631 df-pfx 14661 df-splice 14740 df-reverse 14749 df-s2 14839 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-efmnd 18828 df-grp 18900 df-minusg 18901 df-mulg 19031 df-subg 19085 df-ghm 19175 df-gim 19220 df-cntz 19275 df-oppg 19304 df-symg 19329 df-pmtr 19404 df-psgn 19453 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-drng 20633 df-sra 21065 df-rgmod 21066 df-cnfld 21287 df-zring 21380 df-zrh 21436 df-dsmm 21673 df-frlm 21688 df-mat 22328 df-mdet 22507

This theorem is referenced by: mdetcl 22518 mdetr0 22527 mdetero 22532 mdetuni0 22543 mdetmul 22545 maduf 22563 madurid 22566 madulid 22567 matunit 22600 cramerimp 22608

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