mply1topmatcllem - Metamath Proof Explorer

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Theorem mply1topmatcllem 22832

Description: Lemma for mply1topmatcl 22834. (Contributed by AV, 6-Oct-2019.)

**Hypotheses**
Ref	Expression
mply1topmat.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mply1topmat.q	⊢ 𝑄 = (Poly₁‘𝐴)
mply1topmat.l	⊢ 𝐿 = (Base‘𝑄)
mply1topmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
mply1topmat.m	⊢ · = ( ·_𝑠 ‘𝑃)
mply1topmat.e	⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
mply1topmat.y	⊢ 𝑌 = (var₁‘𝑅)

**Assertion**
Ref	Expression
mply1topmatcllem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝐼((coe₁‘𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0_g‘𝑃))

Distinct variable groups: 𝑘,𝑂 · ,𝑘 𝑘,𝐼 𝑘,𝐽 𝑘,𝐿 𝑘,𝑁 𝑃,𝑘 𝑅,𝑘 Allowed substitution hints: 𝐴(𝑘) 𝑄(𝑘) 𝐸(𝑘) 𝑌(𝑘)

**Proof of Theorem mply1topmatcllem**
Step	Hyp	Ref	Expression
1		nn0ex 12473	. . 3 ⊢ ℕ₀ ∈ V
2	1	a1i 11	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ℕ₀ ∈ V)
3		mply1topmat.p	. . . . 5 ⊢ 𝑃 = (Poly₁‘𝑅)
4	3	ply1lmod 22282	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
5	4	3ad2ant2 1143	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ LMod)
6	5	3ad2ant1 1142	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑃 ∈ LMod)
7		simp12 1214	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑅 ∈ Ring)
8	3	ply1sca 22283	. . 3 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
9	7, 8	syl 17	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑅 = (Scalar‘𝑃))
10		eqid 2752	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
11		ovexd 7416	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝐼((coe₁‘𝑂)‘𝑘)𝐽) ∈ V)
12		eqid 2752	. . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
13	12, 10	mgpbas 20163	. . 3 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
14		mply1topmat.e	. . 3 ⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
15	3	ply1ring 22278	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
16	12	ringmgp 20257	. . . . . . 7 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
18	17	3ad2ant2 1143	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (mulGrp‘𝑃) ∈ Mnd)
19	18	3ad2ant1 1142	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (mulGrp‘𝑃) ∈ Mnd)
20	19	adantr 483	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (mulGrp‘𝑃) ∈ Mnd)
21		simpr 487	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
22		mply1topmat.y	. . . . . . 7 ⊢ 𝑌 = (var₁‘𝑅)
23	22, 3, 10	vr1cl 22248	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
24	23	3ad2ant2 1143	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑌 ∈ (Base‘𝑃))
25	24	3ad2ant1 1142	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑌 ∈ (Base‘𝑃))
26	25	adantr 483	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑌 ∈ (Base‘𝑃))
27	13, 14, 20, 21, 26	mulgnn0cld 19109	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝑘𝐸𝑌) ∈ (Base‘𝑃))
28		eqid 2752	. 2 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
29		eqid 2752	. 2 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
30		mply1topmat.m	. 2 ⊢ · = ( ·_𝑠 ‘𝑃)
31		mply1topmat.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
32		mply1topmat.q	. . 3 ⊢ 𝑄 = (Poly₁‘𝐴)
33		mply1topmat.l	. . 3 ⊢ 𝐿 = (Base‘𝑄)
34	31, 32, 33	mptcoe1matfsupp 22831	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ (𝐼((coe₁‘𝑂)‘𝑘)𝐽)) finSupp (0_g‘𝑅))
35	2, 6, 9, 10, 11, 27, 28, 29, 30, 34	mptscmfsupp0 20963	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝐼((coe₁‘𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0_g‘𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1095 = wceq 1550 ∈ wcel 2132 Vcvv 3444 class class class wbr 5090 ↦ cmpt 5171 ‘cfv 6506 (class class class)co 7381 Fincfn 8912 finSupp cfsupp 9293 ℕ₀cn0 12467 Basecbs 17217 Scalarcsca 17261 ·_𝑠 cvsca 17262 0_gc0g 17440 Mndcmnd 18740 ._gcmg 19081 mulGrpcmgp 20158 Ringcrg 20251 LModclmod 20896 var₁cv1 22207 Poly₁cpl1 22208 coe₁cco1 22209 Mat cmat 22436

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-ot 4581 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-ofr 7646 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-sup 9374 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-fz 13499 df-fzo 13646 df-seq 14001 df-hash 14330 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-hom 17282 df-cco 17283 df-0g 17442 df-gsum 17443 df-prds 17448 df-pws 17450 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-mhm 18789 df-submnd 18790 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-ghm 19226 df-cntz 19329 df-cmn 19794 df-abl 19795 df-mgp 20159 df-rng 20171 df-ur 20200 df-ring 20253 df-subrng 20564 df-subrg 20588 df-lmod 20898 df-lss 20968 df-sra 21209 df-rgmod 21210 df-dsmm 21753 df-frlm 21768 df-psr 21930 df-mvr 21931 df-mpl 21932 df-opsr 21934 df-psr1 22211 df-vr1 22212 df-ply1 22213 df-coe1 22214 df-mat 22437

This theorem is referenced by: mply1topmatcl 22834 mp2pm2mplem2 22836

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