Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cpmatel2 | Structured version Visualization version GIF version |
Description: Another property of a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.) (Proof shortened by AV, 27-Nov-2019.) |
Ref | Expression |
---|---|
cpmat.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
cpmat.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cpmat.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
cpmat.b | ⊢ 𝐵 = (Base‘𝐶) |
cpmatel2.k | ⊢ 𝐾 = (Base‘𝑅) |
cpmatel2.a | ⊢ 𝐴 = (algSc‘𝑃) |
Ref | Expression |
---|---|
cpmatel2 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmat.s | . . 3 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
2 | cpmat.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | cpmat.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
4 | cpmat.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 1, 2, 3, 4 | cpmatel 21562 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑙) = (0g‘𝑅))) |
6 | simpl2 1194 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
7 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
8 | simprl 771 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁) | |
9 | simprr 773 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁) | |
10 | simpl3 1195 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ 𝐵) | |
11 | 3, 7, 4, 8, 9, 10 | matecld 21277 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃)) |
12 | cpmatel2.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
13 | eqid 2736 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | cpmatel2.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑃) | |
15 | 12, 13, 2, 7, 14 | cply1coe0bi 21175 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑃)) → (∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘) ↔ ∀𝑙 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑙) = (0g‘𝑅))) |
16 | 6, 11, 15 | syl2anc 587 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘) ↔ ∀𝑙 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑙) = (0g‘𝑅))) |
17 | 16 | bicomd 226 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑙 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑙) = (0g‘𝑅) ↔ ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘))) |
18 | 17 | 2ralbidva 3109 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑙) = (0g‘𝑅) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘))) |
19 | 5, 18 | bitrd 282 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 ℕcn 11795 Basecbs 16666 0gc0g 16898 Ringcrg 19516 algSccascl 20768 Poly1cpl1 21052 coe1cco1 21053 Mat cmat 21258 ConstPolyMat ccpmat 21554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-ot 4536 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-srg 19475 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-sra 20163 df-rgmod 20164 df-dsmm 20648 df-frlm 20663 df-ascl 20771 df-psr 20822 df-mvr 20823 df-mpl 20824 df-opsr 20826 df-psr1 21055 df-vr1 21056 df-ply1 21057 df-coe1 21058 df-mat 21259 df-cpmat 21557 |
This theorem is referenced by: cpmatelimp2 21565 cpmatacl 21567 cpmatinvcl 21568 |
Copyright terms: Public domain | W3C validator |