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Theorem mp2pm2mplem5 21418

Description: Lemma 5 for mp2pm2mp 21419. (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 11904	. . 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 21052	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
5		mp2pm2mp.q	. . . . 5 ⊢ 𝑄 = (Poly₁‘𝐴)
6	5	ply1lmod 20420	. . . 4 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
7	4, 6	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
8	7	3adant3 1128	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ LMod)
9	4	3adant3 1128	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
10	5	ply1sca 20421	. . 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 1188	. . 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 2821	. . . . 5 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
20		eqid 2821	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
21	3, 5, 12, 14, 15, 16, 17, 18, 19, 20	mply1topmatcl 21413	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
22	21	adantr 483	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
23		simpr 487	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
24		eqid 2821	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
25	14, 19, 20, 3, 24	decpmatcl 21375	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
26	13, 22, 23, 25	syl3anc 1367	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
27		mp2pm2mplem5.x	. . . 4 ⊢ 𝑋 = (var₁‘𝐴)
28		eqid 2821	. . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29		mp2pm2mplem5.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
30	5, 27, 28, 29, 12	ply1moncl 20439	. . 3 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
31	9, 30	sylan 582	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
32		eqid 2821	. 2 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
33		eqid 2821	. 2 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
34		mp2pm2mplem5.m	. 2 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
35		fveq2 6670	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → ((coe₁‘𝑝)‘𝑘) = ((coe₁‘𝑝)‘𝑙))
36	35	oveqd 7173	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑖((coe₁‘𝑝)‘𝑘)𝑗) = (𝑖((coe₁‘𝑝)‘𝑙)𝑗))
37		oveq1 7163	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘𝐸𝑌) = (𝑙𝐸𝑌))
38	36, 37	oveq12d 7174	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
39	38	cbvmptv 5169	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
40	39	oveq2i 7167	. . . . . . . . 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 7232	. . . . . . 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 5158	. . . . . 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 2844	. . . . 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 21417	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) = ((coe₁‘𝑂)‘𝑘))
46	45	mpteq2dva 5161	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((𝐼‘𝑂) decompPMat 𝑘)) = (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)))
47	5, 12, 33	mptcoe1fsupp 20383	. . . 4 ⊢ ((𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
48	4, 47	stoic3 1777	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
49	46, 48	eqbrtrd 5088	. 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 19699	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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Fincfn 8509 finSupp cfsupp 8833 ℕ₀cn0 11898 Basecbs 16483 Scalarcsca 16568 ·_𝑠 cvsca 16569 0_gc0g 16713 Σ_g cgsu 16714 ._gcmg 18224 mulGrpcmgp 19239 Ringcrg 19297 LModclmod 19634 var₁cv1 20344 Poly₁cpl1 20345 coe₁cco1 20346 Mat cmat 21016 decompPMat cdecpmat 21370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-psr 20136 df-mvr 20137 df-mpl 20138 df-opsr 20140 df-psr1 20348 df-vr1 20349 df-ply1 20350 df-coe1 20351 df-dsmm 20876 df-frlm 20891 df-mamu 20995 df-mat 21017 df-decpmat 21371

This theorem is referenced by: mp2pm2mp 21419

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