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Theorem mhmcompl 22318

Description: The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 7-Feb-2025.)

**Hypotheses**
Ref	Expression
mhmcompl.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mhmcompl.q	⊢ 𝑄 = (𝐼 mPoly 𝑆)
mhmcompl.b	⊢ 𝐵 = (Base‘𝑃)
mhmcompl.c	⊢ 𝐶 = (Base‘𝑄)
mhmcompl.h	⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
mhmcompl.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)

**Assertion**
Ref	Expression
mhmcompl	⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)

Proof of Theorem mhmcompl Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6891	. . . 4 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
2		eqid 2735	. . . . 5 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
3		ovexd 7440	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
4	2, 3	rabexd 5310	. . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∈ V)
5		mhmcompl.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
6		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	6, 7	mhmf 18767	. . . . . 6 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
9	5, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
10		mhmcompl.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
11		mhmcompl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
12		mhmcompl.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13	10, 6, 11, 2, 12	mplelf 21958	. . . . 5 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
14	9, 13	fcod 6731	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆))
15	1, 4, 14	elmapdd 8855	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Base‘𝑆) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
16		eqid 2735	. . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
17		eqid 2735	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
18	10, 11	mplrcl 21954	. . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
19	12, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
20	16, 7, 2, 17, 19	psrbas 21893	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
21	15, 20	eleqtrrd 2837	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)))
22		fvexd 6891	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
23		mhmrcl1 18765	. . . . 5 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
24	5, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
25		eqid 2735	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
26	6, 25	mndidcl 18727	. . . 4 ⊢ (𝑅 ∈ Mnd → (0_g‘𝑅) ∈ (Base‘𝑅))
27	24, 26	syl 17	. . 3 ⊢ (𝜑 → (0_g‘𝑅) ∈ (Base‘𝑅))
28		ssidd 3982	. . 3 ⊢ (𝜑 → (Base‘𝑅) ⊆ (Base‘𝑅))
29		fvexd 6891	. . 3 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
30	10, 11, 25, 12	mplelsfi 21955	. . 3 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
31		eqid 2735	. . . . 5 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
32	25, 31	mhm0 18772	. . . 4 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → (𝐻‘(0_g‘𝑅)) = (0_g‘𝑆))
33	5, 32	syl 17	. . 3 ⊢ (𝜑 → (𝐻‘(0_g‘𝑅)) = (0_g‘𝑆))
34	22, 27, 13, 9, 28, 4, 29, 30, 33	fsuppcor 9416	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) finSupp (0_g‘𝑆))
35		mhmcompl.q	. . 3 ⊢ 𝑄 = (𝐼 mPoly 𝑆)
36		mhmcompl.c	. . 3 ⊢ 𝐶 = (Base‘𝑄)
37	35, 16, 17, 31, 36	mplelbas 21951	. 2 ⊢ ((𝐻 ∘ 𝐹) ∈ 𝐶 ↔ ((𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)) ∧ (𝐻 ∘ 𝐹) finSupp (0_g‘𝑆)))
38	21, 34, 37	sylanbrc 583	1 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3415 Vcvv 3459 class class class wbr 5119 ^◡ccnv 5653 “ cima 5657 ∘ ccom 5658 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ↑_m cmap 8840 Fincfn 8959 finSupp cfsupp 9373 ℕcn 12240 ℕ₀cn0 12501 Basecbs 17228 0_gc0g 17453 Mndcmnd 18712 MndHom cmhm 18759 mPwSer cmps 21864 mPoly cmpl 21866

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-tset 17290 df-0g 17455 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-psr 21869 df-mpl 21871

This theorem is referenced by: mhmcoaddmpl 22319 rhmcomulmpl 22320 rhmmpl 22321 mhmcoply1 22323 selvcllem4 42604 selvvvval 42608 selvadd 42611 selvmul 42612

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