mply1topmatcllem - Metamath Proof Explorer

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Theorem mply1topmatcllem 22749

Description: Lemma for mply1topmatcl 22751. (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 12409	. . 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 22194	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
5	4	3ad2ant2 1135	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑃 ∈ LMod)
6	5	3ad2ant1 1134	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑃 ∈ LMod)
7		simp12 1206	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑅 ∈ Ring)
8	3	ply1sca 22195	. . 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 2735	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
11		ovexd 7393	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝐼((coe₁‘𝑂)‘𝑘)𝐽) ∈ V)
12		eqid 2735	. . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
13	12, 10	mgpbas 20082	. . 3 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
14		mply1topmat.e	. . 3 ⊢ 𝐸 = (._g‘(mulGrp‘𝑃))
15	3	ply1ring 22190	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
16	12	ringmgp 20176	. . . . . . 7 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
18	17	3ad2ant2 1135	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (mulGrp‘𝑃) ∈ Mnd)
19	18	3ad2ant1 1134	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (mulGrp‘𝑃) ∈ Mnd)
20	19	adantr 480	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (mulGrp‘𝑃) ∈ Mnd)
21		simpr 484	. . 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 22160	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
24	23	3ad2ant2 1135	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → 𝑌 ∈ (Base‘𝑃))
25	24	3ad2ant1 1134	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → 𝑌 ∈ (Base‘𝑃))
26	25	adantr 480	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → 𝑌 ∈ (Base‘𝑃))
27	13, 14, 20, 21, 26	mulgnn0cld 19027	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) ∧ 𝑘 ∈ ℕ₀) → (𝑘𝐸𝑌) ∈ (Base‘𝑃))
28		eqid 2735	. 2 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
29		eqid 2735	. 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 22748	. 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 20880	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3439 class class class wbr 5097 ↦ cmpt 5178 ‘cfv 6491 (class class class)co 7358 Fincfn 8885 finSupp cfsupp 9266 ℕ₀cn0 12403 Basecbs 17138 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17361 Mndcmnd 18661 ._gcmg 18999 mulGrpcmgp 20077 Ringcrg 20170 LModclmod 20813 var₁cv1 22118 Poly₁cpl1 22119 coe₁cco1 22120 Mat cmat 22353

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 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-sup 9347 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-fz 13426 df-fzo 13573 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19144 df-cntz 19248 df-cmn 19713 df-abl 19714 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-subrng 20481 df-subrg 20505 df-lmod 20815 df-lss 20885 df-sra 21127 df-rgmod 21128 df-dsmm 21689 df-frlm 21704 df-psr 21867 df-mvr 21868 df-mpl 21869 df-opsr 21871 df-psr1 22122 df-vr1 22123 df-ply1 22124 df-coe1 22125 df-mat 22354

This theorem is referenced by: mply1topmatcl 22751 mp2pm2mplem2 22753

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