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Mirrors > Home > MPE Home > Th. List > madetsmelbas2 | Structured version Visualization version GIF version |
Description: A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.) |
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) |
Ref | Expression |
---|---|
madetsmelbas2 | ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 20146 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | 1 | 3ad2ant1 1132 | . 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 22232 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
6 | 5 | simpld 494 | . . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
7 | 6 | 3ad2ant2 1133 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑁 ∈ Fin) |
8 | simp3 1137 | . . 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 21458 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅)) |
13 | 2, 7, 8, 12 | syl3anc 1370 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅)) |
14 | madetsmelbas.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
15 | eqid 2731 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | 14, 15 | mgpbas 20041 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝐺) |
17 | 14 | crngmgp 20142 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
18 | 17 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝐺 ∈ CMnd) |
19 | simp2 1136 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → 𝑀 ∈ 𝐵) | |
20 | 3, 4, 9 | matepm2cl 22285 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 (𝑛𝑀(𝑄‘𝑛)) ∈ (Base‘𝑅)) |
21 | 2, 8, 19, 20 | syl3anc 1370 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → ∀𝑛 ∈ 𝑁 (𝑛𝑀(𝑄‘𝑛)) ∈ (Base‘𝑅)) |
22 | 16, 18, 7, 21 | gsummptcl 19883 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛)))) ∈ (Base‘𝑅)) |
23 | eqid 2731 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
24 | 15, 23 | ringcl 20151 | . 2 ⊢ ((𝑅 ∈ Ring ∧ ((𝑌 ∘ 𝑆)‘𝑄) ∈ (Base‘𝑅) ∧ (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛)))) ∈ (Base‘𝑅)) → (((𝑌 ∘ 𝑆)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (Base‘𝑅)) |
25 | 2, 13, 22, 24 | syl3anc 1370 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ↦ cmpt 5231 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 Fincfn 8945 Basecbs 17151 .rcmulr 17205 Σg cgsu 17393 SymGrpcsymg 19282 pmSgncpsgn 19405 CMndccmn 19696 mulGrpcmgp 20035 Ringcrg 20134 CRingccrg 20135 ℤRHomczrh 21359 Mat cmat 22227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-xnn0 12552 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-word 14472 df-lsw 14520 df-concat 14528 df-s1 14553 df-substr 14598 df-pfx 14628 df-splice 14707 df-reverse 14716 df-s2 14806 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-efmnd 18792 df-grp 18864 df-minusg 18865 df-mulg 18994 df-subg 19046 df-ghm 19135 df-gim 19180 df-cntz 19229 df-oppg 19258 df-symg 19283 df-pmtr 19358 df-psgn 19407 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-sra 21019 df-rgmod 21020 df-cnfld 21234 df-zring 21307 df-zrh 21363 df-dsmm 21597 df-frlm 21612 df-mat 22228 |
This theorem is referenced by: smadiadetlem1 22484 smadiadetlem3lem0 22487 smadiadet 22492 |
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