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Theorem pm2mpf1lem 22841

Description: Lemma for pm2mpf1 22846. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 4-Dec-2019.)

**Hypotheses**
Ref	Expression
pm2mpf1lem.p	⊢ 𝑃 = (Poly₁‘𝑅)
pm2mpf1lem.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pm2mpf1lem.b	⊢ 𝐵 = (Base‘𝐶)
pm2mpf1lem.m	⊢ ∗ = ( ·_𝑠 ‘𝑄)
pm2mpf1lem.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
pm2mpf1lem.x	⊢ 𝑋 = (var₁‘𝐴)
pm2mpf1lem.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pm2mpf1lem.q	⊢ 𝑄 = (Poly₁‘𝐴)

**Assertion**
Ref	Expression
pm2mpf1lem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝐾 𝑘,𝑁 𝑄,𝑘 𝑅,𝑘 𝑈,𝑘 ∗ ,𝑘 ↑ ,𝑘 Allowed substitution hints: 𝐶(𝑘) 𝑃(𝑘) 𝑋(𝑘)

**Proof of Theorem pm2mpf1lem**
Step	Hyp	Ref	Expression
1		pm2mpf1lem.q	. . 3 ⊢ 𝑄 = (Poly₁‘𝐴)
2		eqid 2761	. . 3 ⊢ (Base‘𝑄) = (Base‘𝑄)
3		pm2mpf1lem.x	. . 3 ⊢ 𝑋 = (var₁‘𝐴)
4		pm2mpf1lem.e	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑄))
5		pm2mpf1lem.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 22490	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
7	6	adantr 484	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝐴 ∈ Ring)
8		eqid 2761	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
9		pm2mpf1lem.m	. . 3 ⊢ ∗ = ( ·_𝑠 ‘𝑄)
10		eqid 2761	. . 3 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
11		simpllr 785	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
12		simplrl 786	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑈 ∈ 𝐵)
13		simpr 488	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
14		pm2mpf1lem.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
15		pm2mpf1lem.c	. . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃)
16		pm2mpf1lem.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
17	14, 15, 16, 5, 8	decpmatcl 22814	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
18	11, 12, 13, 17	syl3anc 1389	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) ∧ 𝑘 ∈ ℕ₀) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
19	18	ralrimiva 3153	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ∀𝑘 ∈ ℕ₀ (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
20	14, 15, 16, 5, 10	decpmatfsupp 22816	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑈 decompPMat 𝑘)) finSupp (0_g‘𝐴))
21	20	ad2ant2lr 758	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → (𝑘 ∈ ℕ₀ ↦ (𝑈 decompPMat 𝑘)) finSupp (0_g‘𝐴))
22		simprr 782	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → 𝐾 ∈ ℕ₀)
23	1, 2, 3, 4, 7, 8, 9, 10, 19, 21, 22	gsummoncoe1 22358	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = ⦋𝐾 / 𝑘⦌(𝑈 decompPMat 𝑘))
24		csbov2g 7438	. . 3 ⊢ (𝐾 ∈ ℕ₀ → ⦋𝐾 / 𝑘⦌(𝑈 decompPMat 𝑘) = (𝑈 decompPMat ⦋𝐾 / 𝑘⦌𝑘))
25	24	ad2antll 739	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ⦋𝐾 / 𝑘⦌(𝑈 decompPMat 𝑘) = (𝑈 decompPMat ⦋𝐾 / 𝑘⦌𝑘))
26		csbvarg 4385	. . . 4 ⊢ (𝐾 ∈ ℕ₀ → ⦋𝐾 / 𝑘⦌𝑘 = 𝐾)
27	26	ad2antll 739	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ⦋𝐾 / 𝑘⦌𝑘 = 𝐾)
28	27	oveq2d 7406	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → (𝑈 decompPMat ⦋𝐾 / 𝑘⦌𝑘) = (𝑈 decompPMat 𝐾))
29	23, 25, 28	3eqtrd 2800	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ⦋csb 3850 class class class wbr 5097 ↦ cmpt 5178 ‘cfv 6515 (class class class)co 7390 Fincfn 8920 finSupp cfsupp 9300 ℕ₀cn0 12474 Basecbs 17235 ·_𝑠 cvsca 17280 0_gc0g 17458 Σ_g cgsu 17459 ._gcmg 19099 mulGrpcmgp 20176 Ringcrg 20269 var₁cv1 22225 Poly₁cpl1 22226 coe₁cco1 22227 Mat cmat 22454 decompPMat cdecpmat 22809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-ofr 7655 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-fz 13506 df-fzo 13653 df-seq 14008 df-hash 14337 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-grp 18968 df-minusg 18969 df-sbg 18970 df-mulg 19100 df-subg 19155 df-ghm 19244 df-cntz 19347 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-subrng 20582 df-subrg 20606 df-lmod 20916 df-lss 20986 df-sra 21227 df-rgmod 21228 df-dsmm 21771 df-frlm 21786 df-psr 21948 df-mvr 21949 df-mpl 21950 df-opsr 21952 df-psr1 22229 df-vr1 22230 df-ply1 22231 df-coe1 22232 df-mamu 22438 df-mat 22455 df-decpmat 22810

This theorem is referenced by: pm2mpf1 22846

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