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Theorem mp2pm2mplem5 22785

Description: Lemma 5 for mp2pm2mp 22786. (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 12434	. . 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 22418	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
5		mp2pm2mp.q	. . . . 5 ⊢ 𝑄 = (Poly₁‘𝐴)
6	5	ply1lmod 22225	. . . 4 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
7	4, 6	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
8	7	3adant3 1133	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ LMod)
9	4	3adant3 1133	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
10	5	ply1sca 22226	. . 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 1194	. . 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 2737	. . . . 5 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
20		eqid 2737	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
21	3, 5, 12, 14, 15, 16, 17, 18, 19, 20	mply1topmatcl 22780	. . . 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 2737	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
25	14, 19, 20, 3, 24	decpmatcl 22742	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
26	13, 22, 23, 25	syl3anc 1374	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
27		mp2pm2mplem5.x	. . . 4 ⊢ 𝑋 = (var₁‘𝐴)
28		eqid 2737	. . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29		mp2pm2mplem5.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
30	5, 27, 28, 29, 12	ply1moncl 22246	. . 3 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
31	9, 30	sylan 581	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
32		eqid 2737	. 2 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
33		eqid 2737	. 2 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
34		mp2pm2mplem5.m	. 2 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
35		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → ((coe₁‘𝑝)‘𝑘) = ((coe₁‘𝑝)‘𝑙))
36	35	oveqd 7377	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑖((coe₁‘𝑝)‘𝑘)𝑗) = (𝑖((coe₁‘𝑝)‘𝑙)𝑗))
37		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘𝐸𝑌) = (𝑙𝐸𝑌))
38	36, 37	oveq12d 7378	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
39	38	cbvmptv 5190	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
40	39	oveq2i 7371	. . . . . . . . 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 7438	. . . . . . 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 5182	. . . . . 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 2760	. . . . 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 22784	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) = ((coe₁‘𝑂)‘𝑘))
46	45	mpteq2dva 5179	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((𝐼‘𝑂) decompPMat 𝑘)) = (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)))
47	5, 12, 33	mptcoe1fsupp 22189	. . . 4 ⊢ ((𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
48	4, 47	stoic3 1778	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
49	46, 48	eqbrtrd 5108	. 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 20913	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 1542 ∈ wcel 2114 Vcvv 3430 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6492 (class class class)co 7360 ∈ cmpo 7362 Fincfn 8886 finSupp cfsupp 9267 ℕ₀cn0 12428 Basecbs 17170 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 Σ_g cgsu 17394 ._gcmg 19034 mulGrpcmgp 20112 Ringcrg 20205 LModclmod 20846 var₁cv1 22149 Poly₁cpl1 22150 coe₁cco1 22151 Mat cmat 22382 decompPMat cdecpmat 22737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-subrng 20514 df-subrg 20538 df-lmod 20848 df-lss 20918 df-sra 21160 df-rgmod 21161 df-dsmm 21722 df-frlm 21737 df-psr 21899 df-mvr 21900 df-mpl 21901 df-opsr 21903 df-psr1 22153 df-vr1 22154 df-ply1 22155 df-coe1 22156 df-mamu 22366 df-mat 22383 df-decpmat 22738

This theorem is referenced by: mp2pm2mp 22786

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