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Theorem mhmcompl 22154

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 6878	. . . 4 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
2		eqid 2761	. . . . 5 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
3		ovexd 7427	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
4	2, 3	rabexd 5295	. . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∈ V)
5		mhmcompl.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
6		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	6, 7	mhmf 18806	. . . . . 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 22029	. . . . 5 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
14	9, 13	fcod 6713	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆))
15	1, 4, 14	elmapdd 8818	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Base‘𝑆) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
16		eqid 2761	. . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
17		eqid 2761	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
18	10, 11	mplrcl 22025	. . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
19	12, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
20	16, 7, 2, 17, 19	psrbas 21966	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
21	15, 20	eleqtrrd 2864	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)))
22		fvexd 6878	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
23		mhmrcl1 18804	. . . . 5 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
24	5, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
25		eqid 2761	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
26	6, 25	mndidcl 18766	. . . 4 ⊢ (𝑅 ∈ Mnd → (0_g‘𝑅) ∈ (Base‘𝑅))
27	24, 26	syl 17	. . 3 ⊢ (𝜑 → (0_g‘𝑅) ∈ (Base‘𝑅))
28		ssidd 3959	. . 3 ⊢ (𝜑 → (Base‘𝑅) ⊆ (Base‘𝑅))
29		fvexd 6878	. . 3 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
30	10, 11, 25, 12	mplelsfi 22026	. . 3 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
31		eqid 2761	. . . . 5 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
32	25, 31	mhm0 18811	. . . 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 9347	. 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 22022	. 2 ⊢ ((𝐻 ∘ 𝐹) ∈ 𝐶 ↔ ((𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)) ∧ (𝐻 ∘ 𝐹) finSupp (0_g‘𝑆)))
38	21, 34, 37	sylanbrc 592	1 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 {crab 3413 Vcvv 3453 class class class wbr 5099 ^◡ccnv 5644 “ cima 5648 ∘ ccom 5649 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 ↑_m cmap 8803 Fincfn 8923 finSupp cfsupp 9304 ℕcn 12207 ℕ₀cn0 12478 Basecbs 17228 0_gc0g 17451 Mndcmnd 18751 MndHom cmhm 18798 mPwSer cmps 21936 mPoly cmpl 21938

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

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 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-tset 17288 df-0g 17453 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-psr 21941 df-mpl 21943

This theorem is referenced by: mhmcoaddmpl 22156 rhmcomulmpl 22157 selvcllem4 22171 selvvvval 22175 selvadd 22176 selvmul 22177 rhmmpl 22423 mhmcoply1 22425

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