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| Mirrors > Home > MPE Home > Th. List > cpmadumatpolylem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for cpmadumatpoly 21493. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
| Ref | Expression |
|---|---|
| cpmadumatpoly.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| cpmadumatpoly.b | ⊢ 𝐵 = (Base‘𝐴) |
| cpmadumatpoly.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| cpmadumatpoly.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
| cpmadumatpoly.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
| cpmadumatpoly.r | ⊢ × = (.r‘𝑌) |
| cpmadumatpoly.m0 | ⊢ − = (-g‘𝑌) |
| cpmadumatpoly.0 | ⊢ 0 = (0g‘𝑌) |
| cpmadumatpoly.g | ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) |
| cpmadumatpoly.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
| cpmadumatpoly.m1 | ⊢ · = ( ·𝑠 ‘𝑌) |
| cpmadumatpoly.1 | ⊢ 1 = (1r‘𝑌) |
| cpmadumatpoly.z | ⊢ 𝑍 = (var1‘𝑅) |
| cpmadumatpoly.d | ⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) |
| cpmadumatpoly.j | ⊢ 𝐽 = (𝑁 maAdju 𝑃) |
| cpmadumatpoly.w | ⊢ 𝑊 = (Base‘𝑌) |
| cpmadumatpoly.q | ⊢ 𝑄 = (Poly1‘𝐴) |
| cpmadumatpoly.x | ⊢ 𝑋 = (var1‘𝐴) |
| cpmadumatpoly.m2 | ⊢ ∗ = ( ·𝑠 ‘𝑄) |
| cpmadumatpoly.e | ⊢ ↑ = (.g‘(mulGrp‘𝑄)) |
| cpmadumatpoly.u | ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) |
| Ref | Expression |
|---|---|
| cpmadumatpolylem1 | ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1132 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin) | |
| 2 | crngring 19310 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 3 | 2 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
| 4 | 1, 3 | jca 514 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 5 | 4 | ad2antrr 724 | . . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 6 | cpmadumatpoly.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 7 | cpmadumatpoly.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
| 8 | cpmadumatpoly.s | . . . . 5 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
| 9 | cpmadumatpoly.u | . . . . 5 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) | |
| 10 | 6, 7, 8, 9 | cpm2mf 21362 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:𝑆⟶𝐵) |
| 11 | 5, 10 | syl 17 | . . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑈:𝑆⟶𝐵) |
| 12 | cpmadumatpoly.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 13 | cpmadumatpoly.y | . . . . . 6 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
| 14 | cpmadumatpoly.r | . . . . . 6 ⊢ × = (.r‘𝑌) | |
| 15 | cpmadumatpoly.m0 | . . . . . 6 ⊢ − = (-g‘𝑌) | |
| 16 | cpmadumatpoly.0 | . . . . . 6 ⊢ 0 = (0g‘𝑌) | |
| 17 | cpmadumatpoly.t | . . . . . 6 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
| 18 | cpmadumatpoly.g | . . . . . 6 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) | |
| 19 | 6, 7, 12, 13, 14, 15, 16, 17, 18, 8 | chfacfisfcpmat 21465 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆) |
| 20 | 2, 19 | syl3anl2 1409 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆) |
| 21 | 20 | anassrs 470 | . . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐺:ℕ0⟶𝑆) |
| 22 | fco 6533 | . . 3 ⊢ ((𝑈:𝑆⟶𝐵 ∧ 𝐺:ℕ0⟶𝑆) → (𝑈 ∘ 𝐺):ℕ0⟶𝐵) | |
| 23 | 11, 21, 22 | syl2anc 586 | . 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺):ℕ0⟶𝐵) |
| 24 | 7 | fvexi 6686 | . . . 4 ⊢ 𝐵 ∈ V |
| 25 | nn0ex 11906 | . . . 4 ⊢ ℕ0 ∈ V | |
| 26 | 24, 25 | pm3.2i 473 | . . 3 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V) |
| 27 | elmapg 8421 | . . 3 ⊢ ((𝐵 ∈ V ∧ ℕ0 ∈ V) → ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0) ↔ (𝑈 ∘ 𝐺):ℕ0⟶𝐵)) | |
| 28 | 26, 27 | mp1i 13 | . 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0) ↔ (𝑈 ∘ 𝐺):ℕ0⟶𝐵)) |
| 29 | 23, 28 | mpbird 259 | 1 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Fincfn 8511 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 − cmin 10872 ℕcn 11640 ℕ0cn0 11900 ...cfz 12895 Basecbs 16485 .rcmulr 16568 ·𝑠 cvsca 16571 0gc0g 16715 -gcsg 18107 .gcmg 18226 mulGrpcmgp 19241 1rcur 19253 Ringcrg 19299 CRingccrg 19300 var1cv1 20346 Poly1cpl1 20347 Mat cmat 21018 maAdju cmadu 21243 ConstPolyMat ccpmat 21313 matToPolyMat cmat2pmat 21314 cPolyMatToMat ccpmat2mat 21315 |
| 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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-vr1 20351 df-ply1 20352 df-coe1 20353 df-dsmm 20878 df-frlm 20893 df-mamu 20997 df-mat 21019 df-cpmat 21316 df-mat2pmat 21317 df-cpmat2mat 21318 |
| This theorem is referenced by: cpmadumatpoly 21493 chcoeffeqlem 21495 |
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