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Mirrors > Home > MPE Home > Th. List > cpmidpmatlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for cpmidpmat 21480. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) |
Ref | Expression |
---|---|
cpmidgsum.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
cpmidgsum.b | ⊢ 𝐵 = (Base‘𝐴) |
cpmidgsum.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cpmidgsum.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
cpmidgsum.x | ⊢ 𝑋 = (var1‘𝑅) |
cpmidgsum.e | ⊢ ↑ = (.g‘(mulGrp‘𝑃)) |
cpmidgsum.m | ⊢ · = ( ·𝑠 ‘𝑌) |
cpmidgsum.1 | ⊢ 1 = (1r‘𝑌) |
cpmidgsum.u | ⊢ 𝑈 = (algSc‘𝑃) |
cpmidgsum.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
cpmidgsum.k | ⊢ 𝐾 = (𝐶‘𝑀) |
cpmidgsum.h | ⊢ 𝐻 = (𝐾 · 1 ) |
cpmidgsumm2pm.o | ⊢ 𝑂 = (1r‘𝐴) |
cpmidgsumm2pm.m | ⊢ ∗ = ( ·𝑠 ‘𝐴) |
cpmidgsumm2pm.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
cpmidpmat.g | ⊢ 𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) |
Ref | Expression |
---|---|
cpmidpmatlem2 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑m ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1187 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ Fin) | |
2 | crngring 19307 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
3 | 2 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring) |
4 | 3 | adantr 483 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring) |
5 | cpmidgsum.k | . . . . . 6 ⊢ 𝐾 = (𝐶‘𝑀) | |
6 | cpmidgsum.c | . . . . . . 7 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
7 | cpmidgsum.a | . . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
8 | cpmidgsum.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐴) | |
9 | cpmidgsum.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
10 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
11 | 6, 7, 8, 9, 10 | chpmatply1 21439 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃)) |
12 | 5, 11 | eqeltrid 2917 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃)) |
13 | eqid 2821 | . . . . . 6 ⊢ (coe1‘𝐾) = (coe1‘𝐾) | |
14 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 13, 10, 9, 14 | coe1fvalcl 20379 | . . . . 5 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐾)‘𝑘) ∈ (Base‘𝑅)) |
16 | 12, 15 | sylan 582 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐾)‘𝑘) ∈ (Base‘𝑅)) |
17 | 2 | anim2i 618 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
18 | 7 | matring 21051 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
19 | cpmidgsumm2pm.o | . . . . . . . 8 ⊢ 𝑂 = (1r‘𝐴) | |
20 | 8, 19 | ringidcl 19317 | . . . . . . 7 ⊢ (𝐴 ∈ Ring → 𝑂 ∈ 𝐵) |
21 | 17, 18, 20 | 3syl 18 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑂 ∈ 𝐵) |
22 | 21 | 3adant3 1128 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑂 ∈ 𝐵) |
23 | 22 | adantr 483 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑂 ∈ 𝐵) |
24 | cpmidgsumm2pm.m | . . . . 5 ⊢ ∗ = ( ·𝑠 ‘𝐴) | |
25 | 14, 7, 8, 24 | matvscl 21039 | . . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe1‘𝐾)‘𝑘) ∈ (Base‘𝑅) ∧ 𝑂 ∈ 𝐵)) → (((coe1‘𝐾)‘𝑘) ∗ 𝑂) ∈ 𝐵) |
26 | 1, 4, 16, 23, 25 | syl22anc 836 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐾)‘𝑘) ∗ 𝑂) ∈ 𝐵) |
27 | cpmidpmat.g | . . 3 ⊢ 𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑘) ∗ 𝑂)) | |
28 | 26, 27 | fmptd 6877 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺:ℕ0⟶𝐵) |
29 | 8 | fvexi 6683 | . . . 4 ⊢ 𝐵 ∈ V |
30 | nn0ex 11902 | . . . 4 ⊢ ℕ0 ∈ V | |
31 | 29, 30 | pm3.2i 473 | . . 3 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V) |
32 | elmapg 8418 | . . 3 ⊢ ((𝐵 ∈ V ∧ ℕ0 ∈ V) → (𝐺 ∈ (𝐵 ↑m ℕ0) ↔ 𝐺:ℕ0⟶𝐵)) | |
33 | 31, 32 | mp1i 13 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐺 ∈ (𝐵 ↑m ℕ0) ↔ 𝐺:ℕ0⟶𝐵)) |
34 | 28, 33 | mpbird 259 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑m ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ↦ cmpt 5145 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 Fincfn 8508 ℕ0cn0 11896 Basecbs 16482 ·𝑠 cvsca 16568 .gcmg 18223 mulGrpcmgp 19238 1rcur 19250 Ringcrg 19296 CRingccrg 19297 algSccascl 20083 var1cv1 20343 Poly1cpl1 20344 coe1cco1 20345 Mat cmat 21015 matToPolyMat cmat2pmat 21311 CharPlyMat cchpmat 21433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1501 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-xnn0 11967 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-word 13861 df-lsw 13914 df-concat 13922 df-s1 13949 df-substr 14002 df-pfx 14032 df-splice 14111 df-reverse 14120 df-s2 14209 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-efmnd 18033 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-gim 18398 df-cntz 18446 df-oppg 18473 df-symg 18495 df-pmtr 18569 df-psgn 18618 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-rnghom 19466 df-drng 19503 df-subrg 19532 df-lmod 19635 df-lss 19703 df-sra 19943 df-rgmod 19944 df-ascl 20086 df-psr 20135 df-mvr 20136 df-mpl 20137 df-opsr 20139 df-psr1 20347 df-vr1 20348 df-ply1 20349 df-coe1 20350 df-cnfld 20545 df-zring 20617 df-zrh 20650 df-dsmm 20875 df-frlm 20890 df-mamu 20994 df-mat 21016 df-mdet 21193 df-mat2pmat 21314 df-chpmat 21434 |
This theorem is referenced by: cpmidpmat 21480 |
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