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Theorem madetsmelbas 22340

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 20169	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	3ad2ant1 1131	. 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 22286	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6	5	simpld 494	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
7	6	3ad2ant2 1132	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑁 ∈ Fin)
8		simp3 1136	. . 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 21507	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
13	2, 7, 8, 12	syl3anc 1369	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
14		madetsmelbas.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
15		eqid 2727	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
16	14, 15	mgpbas 20064	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
17	14	crngmgp 20165	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
18	17	3ad2ant1 1131	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝐺 ∈ CMnd)
19		simp2 1135	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑀 ∈ 𝐵)
20	3, 4, 9	matepmcl 22338	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
21	2, 8, 19, 20	syl3anc 1369	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
22	16, 18, 7, 21	gsummptcl 19906	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
23		eqid 2727	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
24	15, 23	ringcl 20174	. 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 1369	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 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ↦ cmpt 5225 ∘ ccom 5676 ‘cfv 6542 (class class class)co 7414 Fincfn 8953 Basecbs 17165 ._rcmulr 17219 Σ_g cgsu 17407 SymGrpcsymg 19305 pmSgncpsgn 19428 CMndccmn 19719 mulGrpcmgp 20058 Ringcrg 20157 CRingccrg 20158 ℤRHomczrh 21405 Mat cmat 22281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-addf 11203 ax-mulf 11204

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-map 8836 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-xnn0 12561 df-z 12575 df-dec 12694 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-seq 13985 df-exp 14045 df-hash 14308 df-word 14483 df-lsw 14531 df-concat 14539 df-s1 14564 df-substr 14609 df-pfx 14639 df-splice 14718 df-reverse 14727 df-s2 14817 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-efmnd 18806 df-grp 18878 df-minusg 18879 df-mulg 19008 df-subg 19062 df-ghm 19152 df-gim 19197 df-cntz 19252 df-oppg 19281 df-symg 19306 df-pmtr 19381 df-psgn 19430 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-sra 21040 df-rgmod 21041 df-cnfld 21260 df-zring 21353 df-zrh 21409 df-dsmm 21646 df-frlm 21661 df-mat 22282

This theorem is referenced by: (None)

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