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Theorem madetsmelbas2 21967

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
madetsmelbas2	⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (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 madetsmelbas2**
Step	Hyp	Ref	Expression
1		crngring 20068	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	3ad2ant1 1134	. 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 21912	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
6	5	simpld 496	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
7	6	3ad2ant2 1135	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑁 ∈ Fin)
8		simp3 1139	. . 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 21148	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
13	2, 7, 8, 12	syl3anc 1372	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅))
14		madetsmelbas.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
15		eqid 2733	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
16	14, 15	mgpbas 19993	. . 3 ⊢ (Base‘𝑅) = (Base‘𝐺)
17	14	crngmgp 20064	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
18	17	3ad2ant1 1134	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝐺 ∈ CMnd)
19		simp2 1138	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑀 ∈ 𝐵)
20	3, 4, 9	matepm2cl 21965	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 (𝑛𝑀(𝑄‘𝑛)) ∈ (Base‘𝑅))
21	2, 8, 19, 20	syl3anc 1372	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ∀𝑛 ∈ 𝑁 (𝑛𝑀(𝑄‘𝑛)) ∈ (Base‘𝑅))
22	16, 18, 7, 21	gsummptcl 19835	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛)))) ∈ (Base‘𝑅))
23		eqid 2733	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
24	15, 23	ringcl 20073	. 2 ⊢ ((𝑅 ∈ Ring ∧ ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅) ∧ (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛)))) ∈ (Base‘𝑅)) → (((𝑌 ∘ 𝑆)‘𝑄)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (Base‘𝑅))
25	2, 13, 22, 24	syl3anc 1372	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (Base‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ↦ cmpt 5232 ∘ ccom 5681 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 Basecbs 17144 ._rcmulr 17198 Σ_g cgsu 17386 SymGrpcsymg 19234 pmSgncpsgn 19357 CMndccmn 19648 mulGrpcmgp 19987 Ringcrg 20056 CRingccrg 20057 ℤRHomczrh 21049 Mat cmat 21907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-word 14465 df-lsw 14513 df-concat 14521 df-s1 14546 df-substr 14591 df-pfx 14621 df-splice 14700 df-reverse 14709 df-s2 14799 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-efmnd 18750 df-grp 18822 df-minusg 18823 df-mulg 18951 df-subg 19003 df-ghm 19090 df-gim 19133 df-cntz 19181 df-oppg 19210 df-symg 19235 df-pmtr 19310 df-psgn 19359 df-cmn 19650 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-rnghom 20251 df-subrg 20317 df-sra 20785 df-rgmod 20786 df-cnfld 20945 df-zring 21018 df-zrh 21053 df-dsmm 21287 df-frlm 21302 df-mat 21908

This theorem is referenced by: smadiadetlem1 22164 smadiadetlem3lem0 22167 smadiadet 22172

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