mply1topmatcllem - Metamath Proof Explorer

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Theorem mply1topmatcllem 21672

Description: Lemma for mply1topmatcl 21674. (Contributed by AV, 6-Oct-2019.)

**Hypotheses**
Ref	Expression
mply1topmat.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mply1topmat.q	⊢ 𝑄 = (Poly₁‘𝐴)
mply1topmat.l	⊢ 𝐿 = (Base‘𝑄)
mply1topmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
mply1topmat.m	⊢ · = ( ·_𝑠 ‘𝑃)
mply1topmat.e	⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
mply1topmat.y	⊢ 𝑌 = (var₁‘𝑅)

**Assertion**
Ref	Expression
mply1topmatcllem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝐼((coe₁‘𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0_g‘𝑃))

Distinct variable groups: 𝑘,𝑂 · ,𝑘 𝑘,𝐼 𝑘,𝐽 𝑘,𝐿 𝑘,𝑁 𝑃,𝑘 𝑅,𝑘 Allowed substitution hints: 𝐴(𝑘) 𝑄(𝑘) 𝐸(𝑘) 𝑌(𝑘)

**Proof of Theorem mply1topmatcllem**
Step	Hyp	Ref	Expression
1		nn0ex 12079	. . 3 ⊢ ℕ₀ ∈ V
2	1	a1i 11	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ℕ₀ ∈ V)
3		mply1topmat.p	. . . . 5 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1lmod 21145	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
5	4	3ad2ant2 1136	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ LMod)
6	5	3ad2ant1 1135	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑃 ∈ LMod)
7		simp12 1206	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑅 ∈ Ring)
8	3	ply1sca 21146	. . 3 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
9	7, 8	syl 17	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑅 = (Scalar‘𝑃))
10		eqid 2734	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
11		ovexd 7237	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝐼((coe₁‘𝑂)‘𝑘)𝐽) ∈ V)
12	3	ply1ring 21141	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
13		eqid 2734	. . . . . . . 8 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
14	13	ringmgp 19540	. . . . . . 7 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
15	12, 14	syl 17	. . . . . 6 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
16	15	3ad2ant2 1136	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (mulGrp‘𝑃) ∈ Mnd)
17	16	3ad2ant1 1135	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (mulGrp‘𝑃) ∈ Mnd)
18	17	adantr 484	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (mulGrp‘𝑃) ∈ Mnd)
19		simpr 488	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
20		mply1topmat.y	. . . . . . 7 ⊢ 𝑌 = (var₁‘𝑅)
21	20, 3, 10	vr1cl 21110	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
22	21	3ad2ant2 1136	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑌 ∈ (Base‘𝑃))
23	22	3ad2ant1 1135	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑌 ∈ (Base‘𝑃))
24	23	adantr 484	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑌 ∈ (Base‘𝑃))
25	13, 10	mgpbas 19482	. . . 4 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
26		mply1topmat.e	. . . 4 ⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
27	25, 26	mulgnn0cl 18480	. . 3 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑘 ∈ ℕ₀ ∧ 𝑌 ∈ (Base‘𝑃)) → (𝑘𝐸𝑌) ∈ (Base‘𝑃))
28	18, 19, 24, 27	syl3anc 1373	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝑘𝐸𝑌) ∈ (Base‘𝑃))
29		eqid 2734	. 2 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
30		eqid 2734	. 2 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
31		mply1topmat.m	. 2 ⊢ · = ( ·_𝑠 ‘𝑃)
32		mply1topmat.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
33		mply1topmat.q	. . 3 ⊢ 𝑄 = (Poly₁‘𝐴)
34		mply1topmat.l	. . 3 ⊢ 𝐿 = (Base‘𝑄)
35	32, 33, 34	mptcoe1matfsupp 21671	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ (𝐼((coe₁‘𝑂)‘𝑘)𝐽)) finSupp (0_g‘𝑅))
36	2, 6, 9, 10, 11, 28, 29, 30, 31, 35	mptscmfsupp0 19936	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝐼((coe₁‘𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0_g‘𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3401 class class class wbr 5043 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 finSupp cfsupp 8974 ℕ₀cn0 12073 Basecbs 16684 Scalarcsca 16770 ·_𝑠 cvsca 16771 0_gc0g 16916 Mndcmnd 18145 ._gcmg 18460 mulGrpcmgp 19476 Ringcrg 19534 LModclmod 19871 var₁cv1 21069 Poly₁cpl1 21070 coe₁cco1 21071 Mat cmat 21276

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-ot 4540 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-gsum 16919 df-prds 16924 df-pws 16926 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-subrg 19770 df-lmod 19873 df-lss 19941 df-sra 20181 df-rgmod 20182 df-dsmm 20666 df-frlm 20681 df-psr 20840 df-mvr 20841 df-mpl 20842 df-opsr 20844 df-psr1 21073 df-vr1 21074 df-ply1 21075 df-coe1 21076 df-mat 21277

This theorem is referenced by: mply1topmatcl 21674 mp2pm2mplem2 21676

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