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Theorem mp2pm2mplem5 22837

Description: Lemma 5 for mp2pm2mp 22838. (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 12559	. . 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 22470	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
5		mp2pm2mp.q	. . . . 5 ⊢ 𝑄 = (Poly₁‘𝐴)
6	5	ply1lmod 22274	. . . 4 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
7	4, 6	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
8	7	3adant3 1132	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ LMod)
9	4	3adant3 1132	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
10	5	ply1sca 22275	. . 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 1192	. . 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 2740	. . . . 5 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
20		eqid 2740	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
21	3, 5, 12, 14, 15, 16, 17, 18, 19, 20	mply1topmatcl 22832	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
22	21	adantr 480	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
23		simpr 484	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
24		eqid 2740	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
25	14, 19, 20, 3, 24	decpmatcl 22794	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
26	13, 22, 23, 25	syl3anc 1371	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
27		mp2pm2mplem5.x	. . . 4 ⊢ 𝑋 = (var₁‘𝐴)
28		eqid 2740	. . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29		mp2pm2mplem5.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
30	5, 27, 28, 29, 12	ply1moncl 22295	. . 3 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
31	9, 30	sylan 579	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
32		eqid 2740	. 2 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
33		eqid 2740	. 2 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
34		mp2pm2mplem5.m	. 2 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
35		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → ((coe₁‘𝑝)‘𝑘) = ((coe₁‘𝑝)‘𝑙))
36	35	oveqd 7465	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑖((coe₁‘𝑝)‘𝑘)𝑗) = (𝑖((coe₁‘𝑝)‘𝑙)𝑗))
37		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘𝐸𝑌) = (𝑙𝐸𝑌))
38	36, 37	oveq12d 7466	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
39	38	cbvmptv 5279	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
40	39	oveq2i 7459	. . . . . . . . 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	mpoeq3ia 7528	. . . . . . 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 5271	. . . . . 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 2768	. . . . 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 22836	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) = ((coe₁‘𝑂)‘𝑘))
46	45	mpteq2dva 5266	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((𝐼‘𝑂) decompPMat 𝑘)) = (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)))
47	5, 12, 33	mptcoe1fsupp 22238	. . . 4 ⊢ ((𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
48	4, 47	stoic3 1774	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
49	46, 48	eqbrtrd 5188	. 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 20947	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 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 Vcvv 3488 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 Fincfn 9003 finSupp cfsupp 9431 ℕ₀cn0 12553 Basecbs 17258 Scalarcsca 17314 ·_𝑠 cvsca 17315 0_gc0g 17499 Σ_g cgsu 17500 ._gcmg 19107 mulGrpcmgp 20161 Ringcrg 20260 LModclmod 20880 var₁cv1 22198 Poly₁cpl1 22199 coe₁cco1 22200 Mat cmat 22432 decompPMat cdecpmat 22789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-subrng 20572 df-subrg 20597 df-lmod 20882 df-lss 20953 df-sra 21195 df-rgmod 21196 df-dsmm 21775 df-frlm 21790 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205 df-mamu 22416 df-mat 22433 df-decpmat 22790

This theorem is referenced by: mp2pm2mp 22838

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