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Mirrors > Home > MPE Home > Th. List > mp2pm2mplem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for mp2pm2mp 20664. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
Ref | Expression |
---|---|
mp2pm2mp.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mp2pm2mp.q | ⊢ 𝑄 = (Poly1‘𝐴) |
mp2pm2mp.l | ⊢ 𝐿 = (Base‘𝑄) |
mp2pm2mp.m | ⊢ · = ( ·𝑠 ‘𝑃) |
mp2pm2mp.e | ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) |
mp2pm2mp.y | ⊢ 𝑌 = (var1‘𝑅) |
mp2pm2mp.i | ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) |
mp2pm2mplem2.p | ⊢ 𝑃 = (Poly1‘𝑅) |
mp2pm2mplem2.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
mp2pm2mplem2.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
mp2pm2mplem2 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp2pm2mplem2.c | . 2 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
2 | eqid 2651 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
3 | mp2pm2mplem2.b | . 2 ⊢ 𝐵 = (Base‘𝐶) | |
4 | simp1 1081 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑁 ∈ Fin) | |
5 | mp2pm2mplem2.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | 5 | ply1ring 19666 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
7 | 6 | 3ad2ant2 1103 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ Ring) |
8 | eqid 2651 | . . 3 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
9 | ringcmn 18627 | . . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd) | |
10 | 6, 9 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ CMnd) |
11 | 10 | 3ad2ant2 1103 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ CMnd) |
12 | 11 | 3ad2ant1 1102 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ CMnd) |
13 | nn0ex 11336 | . . . 4 ⊢ ℕ0 ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ℕ0 ∈ V) |
15 | simpl12 1157 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring) | |
16 | mp2pm2mp.a | . . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
17 | eqid 2651 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | eqid 2651 | . . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
19 | simpl2 1085 | . . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ0) → 𝑖 ∈ 𝑁) | |
20 | simpl3 1086 | . . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ0) → 𝑗 ∈ 𝑁) | |
21 | simp13 1113 | . . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑂 ∈ 𝐿) | |
22 | eqid 2651 | . . . . . . . 8 ⊢ (coe1‘𝑂) = (coe1‘𝑂) | |
23 | mp2pm2mp.l | . . . . . . . 8 ⊢ 𝐿 = (Base‘𝑄) | |
24 | mp2pm2mp.q | . . . . . . . 8 ⊢ 𝑄 = (Poly1‘𝐴) | |
25 | 22, 23, 24, 18 | coe1fvalcl 19630 | . . . . . . 7 ⊢ ((𝑂 ∈ 𝐿 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝑂)‘𝑘) ∈ (Base‘𝐴)) |
26 | 21, 25 | sylan 487 | . . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝑂)‘𝑘) ∈ (Base‘𝐴)) |
27 | 16, 17, 18, 19, 20, 26 | matecld 20280 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ0) → (𝑖((coe1‘𝑂)‘𝑘)𝑗) ∈ (Base‘𝑅)) |
28 | simpr 476 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
29 | mp2pm2mp.y | . . . . . 6 ⊢ 𝑌 = (var1‘𝑅) | |
30 | mp2pm2mp.m | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑃) | |
31 | eqid 2651 | . . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
32 | mp2pm2mp.e | . . . . . 6 ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) | |
33 | 17, 5, 29, 30, 31, 32, 2 | ply1tmcl 19690 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑖((coe1‘𝑂)‘𝑘)𝑗) ∈ (Base‘𝑅) ∧ 𝑘 ∈ ℕ0) → ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)) ∈ (Base‘𝑃)) |
34 | 15, 27, 28, 33 | syl3anc 1366 | . . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ0) → ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)) ∈ (Base‘𝑃)) |
35 | eqid 2651 | . . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))) | |
36 | 34, 35 | fmptd 6425 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))):ℕ0⟶(Base‘𝑃)) |
37 | 16, 24, 23, 5, 30, 32, 29 | mply1topmatcllem 20656 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))) finSupp (0g‘𝑃)) |
38 | 2, 8, 12, 14, 36, 37 | gsumcl 18362 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) ∈ (Base‘𝑃)) |
39 | 1, 2, 3, 4, 7, 38 | matbas2d 20277 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 Fincfn 7997 ℕ0cn0 11330 Basecbs 15904 ·𝑠 cvsca 15992 0gc0g 16147 Σg cgsu 16148 .gcmg 17587 CMndccmn 18239 mulGrpcmgp 18535 Ringcrg 18593 var1cv1 19594 Poly1cpl1 19595 coe1cco1 19596 Mat cmat 20261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-ofr 6940 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-sup 8389 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-fzo 12505 df-seq 12842 df-hash 13158 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-hom 16013 df-cco 16014 df-0g 16149 df-gsum 16150 df-prds 16155 df-pws 16157 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-ghm 17705 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-subrg 18826 df-lmod 18913 df-lss 18981 df-sra 19220 df-rgmod 19221 df-psr 19404 df-mvr 19405 df-mpl 19406 df-opsr 19408 df-psr1 19598 df-vr1 19599 df-ply1 19600 df-coe1 19601 df-dsmm 20124 df-frlm 20139 df-mat 20262 |
This theorem is referenced by: mp2pm2mplem3 20661 |
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