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Mirrors > Home > MPE Home > Th. List > smadiadetlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for smadiadet 20295: A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.) |
Ref | Expression |
---|---|
marep01ma.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
marep01ma.b | ⊢ 𝐵 = (Base‘𝐴) |
marep01ma.r | ⊢ 𝑅 ∈ CRing |
marep01ma.0 | ⊢ 0 = (0g‘𝑅) |
marep01ma.1 | ⊢ 1 = (1r‘𝑅) |
smadiadetlem.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
smadiadetlem.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
madetminlem.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
madetminlem.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
madetminlem.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
smadiadetlem1 | ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))) ∈ (Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marep01ma.r | . . 3 ⊢ 𝑅 ∈ CRing | |
2 | 1 | a1i 11 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑃) → 𝑅 ∈ CRing) |
3 | marep01ma.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | marep01ma.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
5 | marep01ma.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
6 | marep01ma.1 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
7 | 3, 4, 1, 5, 6 | marep01ma 20285 | . . 3 ⊢ (𝑀 ∈ 𝐵 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗))) ∈ 𝐵) |
8 | 7 | ad2antrr 758 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑃) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗))) ∈ 𝐵) |
9 | simpr 476 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝑃) | |
10 | smadiadetlem.p | . . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
11 | madetminlem.s | . . 3 ⊢ 𝑆 = (pmSgn‘𝑁) | |
12 | madetminlem.y | . . 3 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
13 | smadiadetlem.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
14 | 10, 11, 12, 3, 4, 13 | madetsmelbas2 20090 | . 2 ⊢ ((𝑅 ∈ CRing ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗))) ∈ 𝐵 ∧ 𝑝 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))) ∈ (Base‘𝑅)) |
15 | 2, 8, 9, 14 | syl3anc 1318 | 1 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))) ∈ (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ifcif 4036 ↦ cmpt 4643 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 .rcmulr 15769 0gc0g 15923 Σg cgsu 15924 SymGrpcsymg 17620 pmSgncpsgn 17732 mulGrpcmgp 18312 1rcur 18324 CRingccrg 18371 ℤRHomczrh 19667 Mat cmat 20032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-xor 1457 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-reverse 13160 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-subg 17414 df-ghm 17481 df-gim 17524 df-cntz 17573 df-oppg 17599 df-symg 17621 df-pmtr 17685 df-psgn 17734 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-rnghom 18538 df-subrg 18601 df-sra 18993 df-rgmod 18994 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-dsmm 19895 df-frlm 19910 df-mat 20033 |
This theorem is referenced by: smadiadet 20295 |
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