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| Mirrors > Home > MPE Home > Th. List > cpmadumatpolylem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for cpmadumatpoly 20507. (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 1054 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin) | |
| 2 | crngring 18381 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 3 | 2 | 3ad2ant2 1076 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
| 4 | 1, 3 | jca 553 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 5 | 4 | ad2antrr 758 | . . . 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 20376 | . . . 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 20479 | . . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶𝑆) |
| 20 | 2, 19 | syl3anl2 1367 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶𝑆) |
| 21 | 20 | anassrs 678 | . . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝐺:ℕ0⟶𝑆) |
| 22 | fco 5971 | . . 3 ⊢ ((𝑈:𝑆⟶𝐵 ∧ 𝐺:ℕ0⟶𝑆) → (𝑈 ∘ 𝐺):ℕ0⟶𝐵) | |
| 23 | 11, 21, 22 | syl2anc 691 | . 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑈 ∘ 𝐺):ℕ0⟶𝐵) |
| 24 | fvex 6113 | . . . . 5 ⊢ (Base‘𝐴) ∈ V | |
| 25 | 7, 24 | eqeltri 2684 | . . . 4 ⊢ 𝐵 ∈ V |
| 26 | nn0ex 11175 | . . . 4 ⊢ ℕ0 ∈ V | |
| 27 | 25, 26 | pm3.2i 470 | . . 3 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V) |
| 28 | elmapg 7757 | . . 3 ⊢ ((𝐵 ∈ V ∧ ℕ0 ∈ V) → ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑𝑚 ℕ0) ↔ (𝑈 ∘ 𝐺):ℕ0⟶𝐵)) | |
| 29 | 27, 28 | 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)) |
| 30 | 23, 29 | mpbird 246 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 ...cfz 12197 Basecbs 15695 .rcmulr 15769 ·𝑠 cvsca 15772 0gc0g 15923 -gcsg 17247 .gcmg 17363 mulGrpcmgp 18312 1rcur 18324 Ringcrg 18370 CRingccrg 18371 var1cv1 19367 Poly1cpl1 19368 Mat cmat 20032 maAdju cmadu 20257 ConstPolyMat ccpmat 20327 matToPolyMat cmat2pmat 20328 cPolyMatToMat ccpmat2mat 20329 |
| 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 |
| This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 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-pm 7747 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-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-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 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-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 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-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-srg 18329 df-ring 18372 df-cring 18373 df-subrg 18601 df-lmod 18688 df-lss 18754 df-sra 18993 df-rgmod 18994 df-ascl 19135 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-vr1 19372 df-ply1 19373 df-coe1 19374 df-dsmm 19895 df-frlm 19910 df-mamu 20009 df-mat 20033 df-cpmat 20330 df-mat2pmat 20331 df-cpmat2mat 20332 |
| This theorem is referenced by: cpmadumatpoly 20507 chcoeffeqlem 20509 |
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