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Theorem mp2pm2mplem5 20813

Description: Lemma 5 for mp2pm2mp 20814. (Contributed by AV, 12-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)
mp2pm2mplem5.m	⊢ ∗ = ( ·_𝑠 ‘𝑄)
mp2pm2mplem5.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
mp2pm2mplem5.x	⊢ 𝑋 = (var₁‘𝐴)

**Assertion**
Ref	Expression
mp2pm2mplem5	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ (((𝐼‘𝑂) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0_g‘𝑄))

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,𝐸,𝑗 𝑖,𝐿,𝑗 𝑘,𝑁 𝑅,𝑖,𝑗 𝑖,𝑌,𝑗 · ,𝑖,𝑗 𝐴,𝑖,𝑗,𝑘 𝑘,𝐸 𝑘,𝑌 ∗ ,𝑘 Allowed substitution hints: 𝐴(𝑝) 𝑄(𝑖,𝑗,𝑘,𝑝) ↑ (𝑖,𝑗,𝑘,𝑝) 𝐼(𝑖,𝑗,𝑘,𝑝) ∗ (𝑖,𝑗,𝑝) 𝑋(𝑖,𝑗,𝑘,𝑝)

Proof of Theorem mp2pm2mplem5 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0ex 11486	. . 3 ⊢ ℕ₀ ∈ V
2	1	a1i 11	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → ℕ₀ ∈ V)
3		mp2pm2mp.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	3	matring 20447	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
5		mp2pm2mp.q	. . . . 5 ⊢ 𝑄 = (Poly₁‘𝐴)
6	5	ply1lmod 19820	. . . 4 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
7	4, 6	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
8	7	3adant3 1127	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ LMod)
9	4	3adant3 1127	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
10	5	ply1sca 19821	. . 3 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
11	9, 10	syl 17	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 = (Scalar‘𝑄))
12		mp2pm2mp.l	. 2 ⊢ 𝐿 = (Base‘𝑄)
13		simpl2 1230	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
14		mp2pm2mplem2.p	. . . . 5 ⊢ 𝑃 = (Poly₁‘𝑅)
15		mp2pm2mp.m	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑃)
16		mp2pm2mp.e	. . . . 5 ⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
17		mp2pm2mp.y	. . . . 5 ⊢ 𝑌 = (var₁‘𝑅)
18		mp2pm2mp.i	. . . . 5 ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
19		eqid 2756	. . . . 5 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
20		eqid 2756	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
21	3, 5, 12, 14, 15, 16, 17, 18, 19, 20	mply1topmatcl 20808	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
22	21	adantr 472	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
23		simpr 479	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
24		eqid 2756	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
25	14, 19, 20, 3, 24	decpmatcl 20770	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
26	13, 22, 23, 25	syl3anc 1477	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
27		mp2pm2mplem5.x	. . . 4 ⊢ 𝑋 = (var₁‘𝐴)
28		eqid 2756	. . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29		mp2pm2mplem5.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
30	5, 27, 28, 29, 12	ply1moncl 19839	. . 3 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
31	9, 30	sylan 489	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
32		eqid 2756	. 2 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
33		eqid 2756	. 2 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
34		mp2pm2mplem5.m	. 2 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
35		fveq2 6348	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → ((coe₁‘𝑝)‘𝑘) = ((coe₁‘𝑝)‘𝑙))
36	35	oveqd 6826	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑖((coe₁‘𝑝)‘𝑘)𝑗) = (𝑖((coe₁‘𝑝)‘𝑙)𝑗))
37		oveq1 6816	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘𝐸𝑌) = (𝑙𝐸𝑌))
38	36, 37	oveq12d 6827	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
39	38	cbvmptv 4898	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
40	39	oveq2i 6820	. . . . . . . . 9 ⊢ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌))))
41	40	a1i 11	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) = (𝑃 Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))))
42	41	mpt2eq3ia 6881	. . . . . . 7 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))))
43	42	mpteq2i 4889	. . . . . 6 ⊢ (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌))))))
44	18, 43	eqtri 2778	. . . . 5 ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌))))))
45	3, 5, 12, 15, 16, 17, 44, 14	mp2pm2mplem4 20812	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) = ((coe₁‘𝑂)‘𝑘))
46	45	mpteq2dva 4892	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((𝐼‘𝑂) decompPMat 𝑘)) = (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)))
47	5, 12, 33	mptcoe1fsupp 19783	. . . 4 ⊢ ((𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
48	4, 47	stoic3 1846	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
49	46, 48	eqbrtrd 4822	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((𝐼‘𝑂) decompPMat 𝑘)) finSupp (0_g‘𝐴))
50	2, 8, 11, 12, 26, 31, 32, 33, 34, 49	mptscmfsupp0 19126	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ (((𝐼‘𝑂) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0_g‘𝑄))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1628 ∈ wcel 2135 Vcvv 3336 class class class wbr 4800 ↦ cmpt 4877 ‘cfv 6045 (class class class)co 6809 ↦ cmpt2 6811 Fincfn 8117 finSupp cfsupp 8436 ℕ₀cn0 11480 Basecbs 16055 Scalarcsca 16142 ·_𝑠 cvsca 16143 0_gc0g 16298 Σ_g cgsu 16299 ._gcmg 17737 mulGrpcmgp 18685 Ringcrg 18743 LModclmod 19061 var₁cv1 19744 Poly₁cpl1 19745 coe₁cco1 19746 Mat cmat 20411 decompPMat cdecpmat 20765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-inf2 8707 ax-cnex 10180 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 ax-pre-mulgt0 10201

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-fal 1634 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-ot 4326 df-uni 4585 df-int 4624 df-iun 4670 df-iin 4671 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-se 5222 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-isom 6054 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-of 7058 df-ofr 7059 df-om 7227 df-1st 7329 df-2nd 7330 df-supp 7460 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-1o 7725 df-2o 7726 df-oadd 7729 df-er 7907 df-map 8021 df-pm 8022 df-ixp 8071 df-en 8118 df-dom 8119 df-sdom 8120 df-fin 8121 df-fsupp 8437 df-sup 8509 df-oi 8576 df-card 8951 df-pnf 10264 df-mnf 10265 df-xr 10266 df-ltxr 10267 df-le 10268 df-sub 10456 df-neg 10457 df-nn 11209 df-2 11267 df-3 11268 df-4 11269 df-5 11270 df-6 11271 df-7 11272 df-8 11273 df-9 11274 df-n0 11481 df-z 11566 df-dec 11682 df-uz 11876 df-fz 12516 df-fzo 12656 df-seq 12992 df-hash 13308 df-struct 16057 df-ndx 16058 df-slot 16059 df-base 16061 df-sets 16062 df-ress 16063 df-plusg 16152 df-mulr 16153 df-sca 16155 df-vsca 16156 df-ip 16157 df-tset 16158 df-ple 16159 df-ds 16162 df-hom 16164 df-cco 16165 df-0g 16300 df-gsum 16301 df-prds 16306 df-pws 16308 df-mre 16444 df-mrc 16445 df-acs 16447 df-mgm 17439 df-sgrp 17481 df-mnd 17492 df-mhm 17532 df-submnd 17533 df-grp 17622 df-minusg 17623 df-sbg 17624 df-mulg 17738 df-subg 17788 df-ghm 17855 df-cntz 17946 df-cmn 18391 df-abl 18392 df-mgp 18686 df-ur 18698 df-ring 18745 df-subrg 18976 df-lmod 19063 df-lss 19131 df-sra 19370 df-rgmod 19371 df-psr 19554 df-mvr 19555 df-mpl 19556 df-opsr 19558 df-psr1 19748 df-vr1 19749 df-ply1 19750 df-coe1 19751 df-dsmm 20274 df-frlm 20289 df-mamu 20388 df-mat 20412 df-decpmat 20766

This theorem is referenced by: mp2pm2mp 20814

W3C validator