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Theorem madetsmelbas 22327

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 20130	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	3ad2ant1 1133	. 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 22275	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6	5	simpld 494	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
7	6	3ad2ant2 1134	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑁 ∈ Fin)
8		simp3 1138	. . 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 21480	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
13	2, 7, 8, 12	syl3anc 1373	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
14		madetsmelbas.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
15		eqid 2729	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
16	14, 15	mgpbas 20030	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
17	14	crngmgp 20126	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
18	17	3ad2ant1 1133	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝐺 ∈ CMnd)
19		simp2 1137	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑀 ∈ 𝐵)
20	3, 4, 9	matepmcl 22325	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
21	2, 8, 19, 20	syl3anc 1373	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅))
22	16, 18, 7, 21	gsummptcl 19873	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
23		eqid 2729	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
24	15, 23	ringcl 20135	. 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 1373	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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3444 ↦ cmpt 5183 ∘ ccom 5635 ‘cfv 6499 (class class class)co 7369 Fincfn 8895 Basecbs 17155 ._rcmulr 17197 Σ_g cgsu 17379 SymGrpcsymg 19275 pmSgncpsgn 19395 CMndccmn 19686 mulGrpcmgp 20025 Ringcrg 20118 CRingccrg 20119 ℤRHomczrh 21385 Mat cmat 22270

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-word 14455 df-lsw 14504 df-concat 14512 df-s1 14537 df-substr 14582 df-pfx 14612 df-splice 14691 df-reverse 14700 df-s2 14790 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17380 df-gsum 17381 df-prds 17386 df-pws 17388 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-submnd 18687 df-efmnd 18772 df-grp 18844 df-minusg 18845 df-mulg 18976 df-subg 19031 df-ghm 19121 df-gim 19167 df-cntz 19225 df-oppg 19254 df-symg 19276 df-pmtr 19348 df-psgn 19397 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-sra 21056 df-rgmod 21057 df-cnfld 21241 df-zring 21333 df-zrh 21389 df-dsmm 21617 df-frlm 21632 df-mat 22271

This theorem is referenced by: (None)

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