mp2pm2mplem2 - Metamath Proof Explorer

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Theorem mp2pm2mplem2 20526

Description: Lemma 2 for mp2pm2mp 20530. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)

**Hypotheses**
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 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
mp2pm2mplem2.p	⊢ 𝑃 = (Poly₁‘𝑅)
mp2pm2mplem2.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
mp2pm2mplem2.b	⊢ 𝐵 = (Base‘𝐶)

**Assertion**
Ref	Expression
mp2pm2mplem2	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)

Distinct variable groups: 𝐸,𝑝 𝐿,𝑝 𝑖,𝑁,𝑗,𝑝 𝑖,𝑂,𝑗,𝑝,𝑘 𝑃,𝑝 𝑅,𝑝 𝑌,𝑝 · ,𝑝 𝑘,𝐿 𝑃,𝑖,𝑗,𝑘 𝑅,𝑘 · ,𝑘 𝑖,𝐿,𝑗 𝑘,𝑁 𝑅,𝑖,𝑗 Allowed substitution hints: 𝐴(𝑖,𝑗,𝑘,𝑝) 𝐵(𝑖,𝑗,𝑘,𝑝) 𝐶(𝑖,𝑗,𝑘,𝑝) 𝑄(𝑖,𝑗,𝑘,𝑝) · (𝑖,𝑗) 𝐸(𝑖,𝑗,𝑘) 𝐼(𝑖,𝑗,𝑘,𝑝) 𝑌(𝑖,𝑗,𝑘)

**Proof of Theorem mp2pm2mplem2**
Step	Hyp	Ref	Expression
1		mp2pm2mplem2.c	. 2 ⊢ 𝐶 = (𝑁 Mat 𝑃)
2		eqid 2626	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		mp2pm2mplem2.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
4		simp1 1059	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑁 ∈ Fin)
5		mp2pm2mplem2.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
6	5	ply1ring 19532	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7	6	3ad2ant2 1081	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ Ring)
8		eqid 2626	. . 3 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
9		ringcmn 18497	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
10	6, 9	syl 17	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ CMnd)
11	10	3ad2ant2 1081	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ CMnd)
12	11	3ad2ant1 1080	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ CMnd)
13		nn0ex 11243	. . . 4 ⊢ ℕ₀ ∈ V
14	13	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ℕ₀ ∈ V)
15		simpl12 1135	. . . . 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 2626	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
18		eqid 2626	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
19		simpl2 1063	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑖 ∈ 𝑁)
20		simpl3 1064	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑗 ∈ 𝑁)
21		simp13 1091	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑂 ∈ 𝐿)
22		eqid 2626	. . . . . . . 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 19496	. . . . . . 7 ⊢ ((𝑂 ∈ 𝐿 ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝑂)‘𝑘) ∈ (Base‘𝐴))
26	21, 25	sylan 488	. . . . . 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 20146	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝑖((coe₁‘𝑂)‘𝑘)𝑗) ∈ (Base‘𝑅))
28		simpr 477	. . . . 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 2626	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
32		mp2pm2mp.e	. . . . . 6 ⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
33	17, 5, 29, 30, 31, 32, 2	ply1tmcl 19556	. . . . 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 1323	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)) ∈ (Base‘𝑃))
35		eqid 2626	. . . 4 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))
36	34, 35	fmptd 6341	. . 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 20522	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))) finSupp (0_g‘𝑃))
38	2, 8, 12, 14, 36, 37	gsumcl 18232	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) ∈ (Base‘𝑃))
39	1, 2, 3, 4, 7, 38	matbas2d 20143	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1992 Vcvv 3191 ↦ cmpt 4678 ‘cfv 5850 (class class class)co 6605 ↦ cmpt2 6607 Fincfn 7900 ℕ₀cn0 11237 Basecbs 15776 ·_𝑠 cvsca 15861 0_gc0g 16016 Σ_g cgsu 16017 ._gcmg 17456 CMndccmn 18109 mulGrpcmgp 18405 Ringcrg 18463 var₁cv1 19460 Poly₁cpl1 19461 coe₁cco1 19462 Mat cmat 20127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-rep 4736 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903 ax-inf2 8483 ax-cnex 9937 ax-resscn 9938 ax-1cn 9939 ax-icn 9940 ax-addcl 9941 ax-addrcl 9942 ax-mulcl 9943 ax-mulrcl 9944 ax-mulcom 9945 ax-addass 9946 ax-mulass 9947 ax-distr 9948 ax-i2m1 9949 ax-1ne0 9950 ax-1rid 9951 ax-rnegex 9952 ax-rrecex 9953 ax-cnre 9954 ax-pre-lttri 9955 ax-pre-lttrn 9956 ax-pre-ltadd 9957 ax-pre-mulgt0 9958

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-fal 1486 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-nel 2900 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3193 df-sbc 3423 df-csb 3520 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-pss 3576 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-tp 4158 df-op 4160 df-ot 4162 df-uni 4408 df-int 4446 df-iun 4492 df-iin 4493 df-br 4619 df-opab 4679 df-mpt 4680 df-tr 4718 df-eprel 4990 df-id 4994 df-po 5000 df-so 5001 df-fr 5038 df-se 5039 df-we 5040 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-pred 5642 df-ord 5688 df-on 5689 df-lim 5690 df-suc 5691 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-f1 5855 df-fo 5856 df-f1o 5857 df-fv 5858 df-isom 5859 df-riota 6566 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-of 6851 df-ofr 6852 df-om 7014 df-1st 7116 df-2nd 7117 df-supp 7242 df-wrecs 7353 df-recs 7414 df-rdg 7452 df-1o 7506 df-2o 7507 df-oadd 7510 df-er 7688 df-map 7805 df-pm 7806 df-ixp 7854 df-en 7901 df-dom 7902 df-sdom 7903 df-fin 7904 df-fsupp 8221 df-sup 8293 df-oi 8360 df-card 8710 df-pnf 10021 df-mnf 10022 df-xr 10023 df-ltxr 10024 df-le 10025 df-sub 10213 df-neg 10214 df-nn 10966 df-2 11024 df-3 11025 df-4 11026 df-5 11027 df-6 11028 df-7 11029 df-8 11030 df-9 11031 df-n0 11238 df-z 11323 df-dec 11438 df-uz 11632 df-fz 12266 df-fzo 12404 df-seq 12739 df-hash 13055 df-struct 15778 df-ndx 15779 df-slot 15780 df-base 15781 df-sets 15782 df-ress 15783 df-plusg 15870 df-mulr 15871 df-sca 15873 df-vsca 15874 df-ip 15875 df-tset 15876 df-ple 15877 df-ds 15880 df-hom 15882 df-cco 15883 df-0g 16018 df-gsum 16019 df-prds 16024 df-pws 16026 df-mre 16162 df-mrc 16163 df-acs 16165 df-mgm 17158 df-sgrp 17200 df-mnd 17211 df-mhm 17251 df-submnd 17252 df-grp 17341 df-minusg 17342 df-sbg 17343 df-mulg 17457 df-subg 17507 df-ghm 17574 df-cntz 17666 df-cmn 18111 df-abl 18112 df-mgp 18406 df-ur 18418 df-ring 18465 df-subrg 18694 df-lmod 18781 df-lss 18847 df-sra 19086 df-rgmod 19087 df-psr 19270 df-mvr 19271 df-mpl 19272 df-opsr 19274 df-psr1 19464 df-vr1 19465 df-ply1 19466 df-coe1 19467 df-dsmm 19990 df-frlm 20005 df-mat 20128

This theorem is referenced by: mp2pm2mplem3 20527

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