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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 22058 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑙) = (0g‘𝑅))) |
6 | simpl2 1192 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
7 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
8 | simprl 769 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁) | |
9 | simprr 771 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁) | |
10 | simpl3 1193 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ 𝐵) | |
11 | 3, 7, 4, 8, 9, 10 | matecld 21773 | . . . . 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 21669 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑃)) → (∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘) ↔ ∀𝑙 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑙) = (0g‘𝑅))) |
16 | 6, 11, 15 | syl2anc 584 | . . . 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 222 | . . 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 3210 | . 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 278 | 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ‘cfv 6496 (class class class)co 7356 Fincfn 8882 ℕcn 12152 Basecbs 17082 0gc0g 17320 Ringcrg 19962 algSccascl 21256 Poly1cpl1 21546 coe1cco1 21547 Mat cmat 21752 ConstPolyMat ccpmat 22050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-ofr 7617 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-sup 9377 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-fzo 13567 df-seq 13906 df-hash 14230 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-hom 17156 df-cco 17157 df-0g 17322 df-gsum 17323 df-prds 17328 df-pws 17330 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-submnd 18601 df-grp 18750 df-minusg 18751 df-sbg 18752 df-mulg 18871 df-subg 18923 df-ghm 19004 df-cntz 19095 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-srg 19916 df-ring 19964 df-subrg 20218 df-lmod 20322 df-lss 20391 df-sra 20631 df-rgmod 20632 df-dsmm 21136 df-frlm 21151 df-ascl 21259 df-psr 21309 df-mvr 21310 df-mpl 21311 df-opsr 21313 df-psr1 21549 df-vr1 21550 df-ply1 21551 df-coe1 21552 df-mat 21753 df-cpmat 22053 |
This theorem is referenced by: cpmatelimp2 22061 cpmatacl 22063 cpmatinvcl 22064 |
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