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Theorem madetsmelbas 20792

Description: A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)

**Hypotheses**
Ref	Expression
madetsmelbas.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
madetsmelbas.s	⊢ 𝑆 = (pmSgn‘𝑁)
madetsmelbas.y	⊢ 𝑌 = (ℤRHom‘𝑅)
madetsmelbas.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
madetsmelbas.b	⊢ 𝐵 = (Base‘𝐴)
madetsmelbas.g	⊢ 𝐺 = (mulGrp‘𝑅)

**Assertion**
Ref	Expression
madetsmelbas	⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑃,𝑛 𝑄,𝑛 𝑅,𝑛 Allowed substitution hints: 𝐴(𝑛) 𝑆(𝑛) 𝐺(𝑛) 𝑌(𝑛)

**Proof of Theorem madetsmelbas**
Step	Hyp	Ref	Expression
1		crngring 19043	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	3ad2ant1 1114	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑅 ∈ Ring)
3		madetsmelbas.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4		madetsmelbas.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
5	3, 4	matrcl 20740	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6	5	simpld 487	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
7	6	3ad2ant2 1115	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑁 ∈ Fin)
8		simp3 1119	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑄 ∈ 𝑃)
9		madetsmelbas.p	. . . 4 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
10		madetsmelbas.s	. . . 4 ⊢ 𝑆 = (pmSgn‘𝑁)
11		madetsmelbas.y	. . . 4 ⊢ 𝑌 = (ℤRHom‘𝑅)
12	9, 10, 11	zrhcopsgnelbas 20456	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
13	2, 7, 8, 12	syl3anc 1352	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
14		madetsmelbas.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
15		eqid 2771	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
16	14, 15	mgpbas 18980	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
17	14	crngmgp 19040	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
18	17	3ad2ant1 1114	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝐺 ∈ CMnd)
19		simp2 1118	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑀 ∈ 𝐵)
20	3, 4, 9	matepmcl 20790	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
21	2, 8, 19, 20	syl3anc 1352	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
22	16, 18, 7, 21	gsummptcl 18852	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
23		eqid 2771	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
24	15, 23	ringcl 19046	. 2 ⊢ ((𝑅 ∈ Ring ∧ ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅) ∧ (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅)) → (((𝑌 ∘ 𝑆)‘𝑄)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
25	2, 13, 22, 24	syl3anc 1352	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∀wral 3081 Vcvv 3408 ↦ cmpt 5004 ∘ ccom 5407 ‘cfv 6185 (class class class)co 6974 Fincfn 8304 Basecbs 16337 ._rcmulr 16420 Σ_g cgsu 16568 SymGrpcsymg 18278 pmSgncpsgn 18390 CMndccmn 18678 mulGrpcmgp 18974 Ringcrg 19032 CRingccrg 19033 ℤRHomczrh 20364 Mat cmat 20735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-addf 10412 ax-mulf 10413

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-xor 1490 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-sup 8699 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-xnn0 11778 df-z 11792 df-dec 11910 df-uz 12057 df-rp 12203 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-word 13671 df-lsw 13724 df-concat 13732 df-s1 13757 df-substr 13802 df-pfx 13851 df-splice 13958 df-reverse 13976 df-s2 14070 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-0g 16569 df-gsum 16570 df-prds 16575 df-pws 16577 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-mulg 18024 df-subg 18072 df-ghm 18139 df-gim 18182 df-cntz 18230 df-oppg 18257 df-symg 18279 df-pmtr 18343 df-psgn 18392 df-cmn 18680 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-rnghom 19202 df-subrg 19268 df-sra 19678 df-rgmod 19679 df-cnfld 20263 df-zring 20335 df-zrh 20368 df-dsmm 20593 df-frlm 20608 df-mat 20736

This theorem is referenced by: (None)

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