mp2pm2mplem2 - Metamath Proof Explorer

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Theorem mp2pm2mplem2 23126

Description: Lemma 2 for mp2pm2mp 23130. (Contributed by AV, 10-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)
mp2pm2mplem2.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
mp2pm2mplem2.b	⊢ 𝐵 = (Base‘𝐶)

**Assertion**
Ref	Expression
mp2pm2mplem2	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)

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

**Proof of Theorem mp2pm2mplem2**
Step	Hyp	Ref	Expression
1		mp2pm2mplem2.c	. 2 ⊢ 𝐶 = (𝑁 Mat 𝑃)
2		eqid 2761	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		mp2pm2mplem2.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
4		simp1 1154	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑁 ∈ Fin)
5		mp2pm2mplem2.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
6	5	ply1ring 22566	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7	6	3ad2ant2 1152	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ Ring)
8		eqid 2761	. . 3 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
9		ringcmn 20512	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
10	6, 9	syl 18	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ CMnd)
11	10	3ad2ant2 1152	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ CMnd)
12	11	3ad2ant1 1151	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ CMnd)
13		nn0ex 12613	. . . 4 ⊢ ℕ₀ ∈ V
14	13	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ℕ₀ ∈ V)
15		simpl12 1268	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
16		mp2pm2mp.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
17		eqid 2761	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
18		eqid 2761	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
19		simpl2 1211	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑖 ∈ 𝑁)
20		simpl3 1212	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑗 ∈ 𝑁)
21		simp13 1224	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑂 ∈ 𝐿)
22		eqid 2761	. . . . . . . 8 ⊢ (coe₁‘𝑂) = (coe₁‘𝑂)
23		mp2pm2mp.l	. . . . . . . 8 ⊢ 𝐿 = (Base‘𝑄)
24		mp2pm2mp.q	. . . . . . . 8 ⊢ 𝑄 = (Poly₁‘𝐴)
25	22, 23, 24, 18	coe1fvalcl 22531	. . . . . . 7 ⊢ ((𝑂 ∈ 𝐿 ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝑂)‘𝑘) ∈ (Base‘𝐴))
26	21, 25	sylan 592	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → ((coe₁‘𝑂)‘𝑘) ∈ (Base‘𝐴))
27	16, 17, 18, 19, 20, 26	matecld 22742	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝑖((coe₁‘𝑂)‘𝑘)𝑗) ∈ (Base‘𝑅))
28		simpr 490	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
29		mp2pm2mp.y	. . . . . 6 ⊢ 𝑌 = (var₁‘𝑅)
30		mp2pm2mp.m	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑃)
31		eqid 2761	. . . . . 6 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
32		mp2pm2mp.e	. . . . . 6 ⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
33	17, 5, 29, 30, 31, 32, 2	ply1tmcl 22592	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑖((coe₁‘𝑂)‘𝑘)𝑗) ∈ (Base‘𝑅) ∧ 𝑘 ∈ ℕ₀) → ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)) ∈ (Base‘𝑃))
34	15, 27, 28, 33	syl3anc 1398	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)) ∈ (Base‘𝑃))
35	34	fmpttd 7116	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))):ℕ₀⟶(Base‘𝑃))
36	16, 24, 23, 5, 30, 32, 29	mply1topmatcllem 23122	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))) finSupp (0_g‘𝑃))
37	2, 8, 12, 14, 35, 36	gsumcl 20130	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))) ∈ (Base‘𝑃))
38	1, 2, 3, 4, 7, 37	matbas2d 22739	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ↦ cmpt 5186 ‘cfv 6539 (class class class)co 7421 ∈ cmpo 7423 Fincfn 8974 ℕ₀cn0 12607 Basecbs 17388 ·_𝑠 cvsca 17433 0_gc0g 17611 Σ_g cgsu 17612 ._gcmg 19278 CMndccmn 19995 mulGrpcmgp 20361 Ringcrg 20460 var₁cv1 22495 Poly₁cpl1 22496 coe₁cco1 22497 Mat cmat 22723

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 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 8179 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-er 8718 df-map 8850 df-pm 8851 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9355 df-sup 9435 df-oi 9505 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-z 12695 df-dec 12816 df-uz 12967 df-fz 13641 df-fzo 13790 df-seq 14146 df-hash 14476 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-ip 17447 df-tset 17448 df-ple 17449 df-ds 17451 df-hom 17453 df-cco 17454 df-0g 17613 df-gsum 17614 df-prds 17619 df-pws 17621 df-mre 17757 df-mrc 17758 df-acs 17760 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-mhm 18979 df-submnd 18980 df-grp 19148 df-minusg 19149 df-sbg 19150 df-mulg 19279 df-subg 19334 df-ghm 19429 df-cntz 19532 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-subrng 20799 df-subrg 20823 df-lmod 21138 df-lss 21208 df-sra 21449 df-rgmod 21450 df-dsmm 22039 df-frlm 22054 df-psr 22218 df-mvr 22219 df-mpl 22220 df-opsr 22222 df-psr1 22499 df-vr1 22500 df-ply1 22501 df-coe1 22502 df-mat 22724

This theorem is used by: mp2pm2mplem3 23127

W3C validator