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Theorem mhmcompl 22324

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 6849	. . . 4 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
2		eqid 2736	. . . . 5 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
3		ovexd 7393	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
4	2, 3	rabexd 5285	. . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∈ V)
5		mhmcompl.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
6		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	6, 7	mhmf 18714	. . . . . 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 21953	. . . . 5 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
14	9, 13	fcod 6687	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆))
15	1, 4, 14	elmapdd 8778	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Base‘𝑆) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
16		eqid 2736	. . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
17		eqid 2736	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
18	10, 11	mplrcl 21949	. . . . 5 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V)
19	12, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
20	16, 7, 2, 17, 19	psrbas 21889	. . 3 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
21	15, 20	eleqtrrd 2839	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (Base‘(𝐼 mPwSer 𝑆)))
22		fvexd 6849	. . 3 ⊢ (𝜑 → (0_g‘𝑆) ∈ V)
23		mhmrcl1 18712	. . . . 5 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
24	5, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
25		eqid 2736	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
26	6, 25	mndidcl 18674	. . . 4 ⊢ (𝑅 ∈ Mnd → (0_g‘𝑅) ∈ (Base‘𝑅))
27	24, 26	syl 17	. . 3 ⊢ (𝜑 → (0_g‘𝑅) ∈ (Base‘𝑅))
28		ssidd 3957	. . 3 ⊢ (𝜑 → (Base‘𝑅) ⊆ (Base‘𝑅))
29		fvexd 6849	. . 3 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
30	10, 11, 25, 12	mplelsfi 21950	. . 3 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑅))
31		eqid 2736	. . . . 5 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
32	25, 31	mhm0 18719	. . . 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 9307	. 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 21946	. 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 1541 ∈ wcel 2113 {crab 3399 Vcvv 3440 class class class wbr 5098 ^◡ccnv 5623 “ cima 5627 ∘ ccom 5628 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ↑_m cmap 8763 Fincfn 8883 finSupp cfsupp 9264 ℕcn 12145 ℕ₀cn0 12401 Basecbs 17136 0_gc0g 17359 Mndcmnd 18659 MndHom cmhm 18706 mPwSer cmps 21860 mPoly cmpl 21862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 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-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-tset 17196 df-0g 17361 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-psr 21865 df-mpl 21867

This theorem is referenced by: mhmcoaddmpl 22325 rhmcomulmpl 22326 rhmmpl 22327 mhmcoply1 22329 selvcllem4 42824 selvvvval 42828 selvadd 42831 selvmul 42832

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