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Theorem mp2pm2mplem5 23134

Description: Lemma 5 for mp2pm2mp 23135. (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 12616	. . 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 22764	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
5		mp2pm2mp.q	. . . . 5 ⊢ 𝑄 = (Poly₁‘𝐴)
6	5	ply1lmod 22575	. . . 4 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
7	4, 6	syl 18	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
8	7	3adant3 1150	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑄 ∈ LMod)
9	4	3adant3 1150	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 ∈ Ring)
10	5	ply1sca 22576	. . 3 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
11	9, 10	syl 18	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝐴 = (Scalar‘𝑄))
12		mp2pm2mp.l	. 2 ⊢ 𝐿 = (Base‘𝑄)
13		simpl2 1211	. . 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 2760	. . . . 5 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
20		eqid 2760	. . . . 5 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
21	3, 5, 12, 14, 15, 16, 17, 18, 19, 20	mply1topmatcl 23129	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
22	21	adantr 486	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)))
23		simpr 490	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
24		eqid 2760	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
25	14, 19, 20, 3, 24	decpmatcl 23091	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑂) ∈ (Base‘(𝑁 Mat 𝑃)) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
26	13, 22, 23, 25	syl3anc 1398	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) ∈ (Base‘𝐴))
27		mp2pm2mplem5.x	. . . 4 ⊢ 𝑋 = (var₁‘𝐴)
28		eqid 2760	. . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
29		mp2pm2mplem5.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
30	5, 27, 28, 29, 12	ply1moncl 22596	. . 3 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
31	9, 30	sylan 592	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐿)
32		eqid 2760	. 2 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
33		eqid 2760	. 2 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
34		mp2pm2mplem5.m	. 2 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
35		fveq2 6886	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑙 → ((coe₁‘𝑝)‘𝑘) = ((coe₁‘𝑝)‘𝑙))
36	35	oveqd 7438	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑖((coe₁‘𝑝)‘𝑘)𝑗) = (𝑖((coe₁‘𝑝)‘𝑙)𝑗))
37		oveq1 7428	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘𝐸𝑌) = (𝑙𝐸𝑌))
38	36, 37	oveq12d 7439	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)) = ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
39	38	cbvmptv 5208	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))) = (𝑙 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑙)𝑗) · (𝑙𝐸𝑌)))
40	39	oveq2i 7432	. . . . . . . . 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 7499	. . . . . . 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 5200	. . . . . 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 2783	. . . . 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 23133	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑘 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝑘) = ((coe₁‘𝑂)‘𝑘))
46	45	mpteq2dva 5197	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((𝐼‘𝑂) decompPMat 𝑘)) = (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)))
47	5, 12, 33	mptcoe1fsupp 22539	. . . 4 ⊢ ((𝐴 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
48	4, 47	stoic3 1808	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ ((coe₁‘𝑂)‘𝑘)) finSupp (0_g‘𝐴))
49	46, 48	eqbrtrd 5126	. 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 21208	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 Vcvv 3450 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6539 (class class class)co 7421 ∈ cmpo 7423 Fincfn 8977 finSupp cfsupp 9357 ℕ₀cn0 12610 Basecbs 17393 Scalarcsca 17437 ·_𝑠 cvsca 17438 0_gc0g 17616 Σ_g cgsu 17617 ._gcmg 19283 mulGrpcmgp 20366 Ringcrg 20465 LModclmod 21141 var₁cv1 22500 Poly₁cpl1 22501 coe₁cco1 22502 Mat cmat 22728 decompPMat cdecpmat 23086

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-ofr 7695 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-sup 9438 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-fz 13644 df-fzo 13793 df-seq 14149 df-hash 14479 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-hom 17458 df-cco 17459 df-0g 17618 df-gsum 17619 df-prds 17624 df-pws 17626 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-mhm 18984 df-submnd 18985 df-grp 19153 df-minusg 19154 df-sbg 19155 df-mulg 19284 df-subg 19339 df-ghm 19434 df-cntz 19537 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-subrng 20804 df-subrg 20828 df-lmod 21143 df-lss 21213 df-sra 21454 df-rgmod 21455 df-dsmm 22044 df-frlm 22059 df-psr 22223 df-mvr 22224 df-mpl 22225 df-opsr 22227 df-psr1 22504 df-vr1 22505 df-ply1 22506 df-coe1 22507 df-mamu 22712 df-mat 22729 df-decpmat 23087

This theorem is used by: mp2pm2mp 23135

W3C validator