mply1topmatcllem - Metamath Proof Explorer

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Theorem mply1topmatcllem 22870

Description: Lemma for mply1topmatcl 22872. (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 12497	. . 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 22320	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
5	4	3ad2ant2 1148	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ LMod)
6	5	3ad2ant1 1147	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑃 ∈ LMod)
7		simp12 1219	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑅 ∈ Ring)
8	3	ply1sca 22321	. . 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 2763	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
11		ovexd 7431	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝐼((coe₁‘𝑂)‘𝑘)𝐽) ∈ V)
12		eqid 2763	. . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
13	12, 10	mgpbas 20201	. . 3 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
14		mply1topmat.e	. . 3 ⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
15	3	ply1ring 22316	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
16	12	ringmgp 20299	. . . . . . 7 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
18	17	3ad2ant2 1148	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (mulGrp‘𝑃) ∈ Mnd)
19	18	3ad2ant1 1147	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (mulGrp‘𝑃) ∈ Mnd)
20	19	adantr 484	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (mulGrp‘𝑃) ∈ Mnd)
21		simpr 488	. . 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 22286	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
24	23	3ad2ant2 1148	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑌 ∈ (Base‘𝑃))
25	24	3ad2ant1 1147	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑌 ∈ (Base‘𝑃))
26	25	adantr 484	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑌 ∈ (Base‘𝑃))
27	13, 14, 20, 21, 26	mulgnn0cld 19147	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝑘𝐸𝑌) ∈ (Base‘𝑃))
28		eqid 2763	. 2 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
29		eqid 2763	. 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 22869	. 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 21001	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 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 Vcvv 3455 class class class wbr 5101 ↦ cmpt 5182 ‘cfv 6521 (class class class)co 7396 Fincfn 8927 finSupp cfsupp 9305 ℕ₀cn0 12491 Basecbs 17255 Scalarcsca 17299 ·_𝑠 cvsca 17300 0_gc0g 17478 Mndcmnd 18778 ._gcmg 19119 mulGrpcmgp 20196 Ringcrg 20293 LModclmod 20934 var₁cv1 22245 Poly₁cpl1 22246 coe₁cco1 22247 Mat cmat 22474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-ot 4592 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-ofr 7661 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9306 df-sup 9386 df-oi 9456 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-2 12290 df-3 12291 df-4 12292 df-5 12293 df-6 12294 df-7 12295 df-8 12296 df-9 12297 df-n0 12492 df-z 12579 df-dec 12699 df-uz 12850 df-fz 13523 df-fzo 13670 df-seq 14025 df-hash 14354 df-struct 17193 df-sets 17210 df-slot 17228 df-ndx 17240 df-base 17256 df-ress 17277 df-plusg 17309 df-mulr 17310 df-sca 17312 df-vsca 17313 df-ip 17314 df-tset 17315 df-ple 17316 df-ds 17318 df-hom 17320 df-cco 17321 df-0g 17480 df-gsum 17481 df-prds 17486 df-pws 17488 df-mre 17624 df-mrc 17625 df-acs 17627 df-mgm 18684 df-sgrp 18763 df-mnd 18779 df-mhm 18827 df-submnd 18828 df-grp 18988 df-minusg 18989 df-sbg 18990 df-mulg 19120 df-subg 19175 df-ghm 19264 df-cntz 19367 df-cmn 19832 df-abl 19833 df-mgp 20197 df-rng 20209 df-ur 20242 df-ring 20295 df-subrng 20606 df-subrg 20630 df-lmod 20936 df-lss 21006 df-sra 21247 df-rgmod 21248 df-dsmm 21791 df-frlm 21806 df-psr 21968 df-mvr 21969 df-mpl 21970 df-opsr 21972 df-psr1 22249 df-vr1 22250 df-ply1 22251 df-coe1 22252 df-mat 22475

This theorem is referenced by: mply1topmatcl 22872 mp2pm2mplem2 22874

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