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Theorem mdetf 22655

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 20295	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	adantr 484	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑅 ∈ Ring)
4		ringcmn 20332	. . . 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 22472	. . . . . 6 ⊢ (𝑚 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
9	8	adantl 485	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
10	9	simpld 498	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → 𝑁 ∈ Fin)
11		eqid 2762	. . . . 5 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
12		eqid 2762	. . . . 5 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
13	11, 12	symgbasfi 19419	. . . 4 ⊢ (𝑁 ∈ Fin → (Base‘(SymGrp‘𝑁)) ∈ Fin)
14	10, 13	syl 17	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → (Base‘(SymGrp‘𝑁)) ∈ Fin)
15	2	ad2antrr 736	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑅 ∈ Ring)
16		zrhpsgnmhm 21636	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
17	3, 10, 16	syl2anc 593	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
18		eqid 2762	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
19	18, 1	mgpbas 20191	. . . . . . . 8 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
20	12, 19	mhmf 18823	. . . . . . 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 7065	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝) ∈ 𝐾)
23	18	crngmgp 20291	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
24	23	ad2antrr 736	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → (mulGrp‘𝑅) ∈ CMnd)
25	10	adantr 484	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑁 ∈ Fin)
26	6, 1, 7	matbas2i 22482	. . . . . . . . . 10 ⊢ (𝑚 ∈ 𝐵 → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
27	26	ad3antlr 741	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑚 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
28		elmapi 8830	. . . . . . . . 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 19416	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Base‘(SymGrp‘𝑁)) → 𝑝:𝑁⟶𝑁)
31	30	adantl 485	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → 𝑝:𝑁⟶𝑁)
32	31	ffvelcdmda 7065	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → (𝑝‘𝑐) ∈ 𝑁)
33		simpr 488	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁)
34	29, 32, 33	fovcdmd 7568	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) ∧ 𝑐 ∈ 𝑁) → ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
35	34	ralrimiva 3154	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ∀𝑐 ∈ 𝑁 ((𝑝‘𝑐)𝑚𝑐) ∈ 𝐾)
36	19, 24, 25, 35	gsummptcl 20007	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ 𝐵) ∧ 𝑝 ∈ (Base‘(SymGrp‘𝑁))) → ((mulGrp‘𝑅) Σ_g (𝑐 ∈ 𝑁 ↦ ((𝑝‘𝑐)𝑚𝑐))) ∈ 𝐾)
37		eqid 2762	. . . . . 6 ⊢ (._r‘𝑅) = (._r‘𝑅)
38	1, 37	ringcl 20300	. . . . 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 1390	. . . 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 3154	. . 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 20007	. 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 2762	. . 3 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
44		eqid 2762	. . 3 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁)
45	42, 6, 7, 12, 43, 44, 37, 18	mdetfval 22646	. 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 7095	1 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 Vcvv 3454 ↦ cmpt 5181 × cxp 5645 ∘ ccom 5651 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ↑_m cmap 8808 Fincfn 8927 Basecbs 17245 ._rcmulr 17287 Σ_g cgsu 17469 MndHom cmhm 18815 SymGrpcsymg 19409 pmSgncpsgn 19529 CMndccmn 19820 mulGrpcmgp 20186 Ringcrg 20283 CRingccrg 20284 ℤRHomczrh 21551 Mat cmat 22467 maDet cmdat 22644

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-addf 11152 ax-mulf 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-xor 1532 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-sup 9388 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-xnn0 12555 df-z 12569 df-dec 12689 df-uz 12840 df-rp 12994 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075 df-hash 14344 df-word 14527 df-lsw 14576 df-concat 14584 df-s1 14610 df-substr 14655 df-pfx 14685 df-splice 14763 df-reverse 14772 df-s2 14861 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-0g 17470 df-gsum 17471 df-prds 17476 df-pws 17478 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-mhm 18817 df-submnd 18818 df-efmnd 18903 df-grp 18978 df-minusg 18979 df-mulg 19110 df-subg 19165 df-ghm 19254 df-gim 19299 df-cntz 19357 df-oppg 19386 df-symg 19410 df-pmtr 19482 df-psgn 19531 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20232 df-ring 20285 df-cring 20286 df-oppr 20386 df-dvdsr 20406 df-unit 20407 df-invr 20437 df-dvr 20450 df-rhm 20521 df-subrng 20596 df-subrg 20620 df-drng 20781 df-sra 21240 df-rgmod 21241 df-cnfld 21425 df-zring 21499 df-zrh 21555 df-dsmm 21784 df-frlm 21799 df-mat 22468 df-mdet 22645

This theorem is referenced by: mdetcl 22656 mdetr0 22665 mdetero 22670 mdetuni0 22681 mdetmul 22683 maduf 22701 madurid 22704 madulid 22705 matunit 22738 cramerimp 22746

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