Theorem rhmply1vsca 22514

Description: Apply a ring homomorphism between two univariate polynomial algebras to a scaled polynomial. (Contributed by SN, 20-May-2025.)

Hypotheses

Ref	Expression
rhmply1vsca.p	⊢ 𝑃 = (Poly1‘𝑅)
rhmply1vsca.q	⊢ 𝑄 = (Poly1‘𝑆)
rhmply1vsca.b	⊢ 𝐵 = (Base‘𝑃)
rhmply1vsca.k	⊢ 𝐾 = (Base‘𝑅)
rhmply1vsca.f	⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
rhmply1vsca.t	⊢ · = ( ·𝑠 ‘𝑃)
rhmply1vsca.u	⊢ ∙ = ( ·𝑠 ‘𝑄)
rhmply1vsca.h	⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
rhmply1vsca.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)
rhmply1vsca.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
rhmply1vsca	⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐹‘𝑋)))

Distinct variable groups:   𝐶,𝑝   𝑋,𝑝   𝐻,𝑝   𝐵,𝑝   · ,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝑃(𝑝)   𝑄(𝑝)   𝑅(𝑝)   𝑆(𝑝)   ∙ (𝑝)   𝐹(𝑝)   𝐾(𝑝)

Proof of Theorem rhmply1vsca

Dummy variables 𝑎 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmply1vsca.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
2		fconst6g 6768	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐾 → ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}):{ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}):{ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
4		psr1baslem 22314	. . . . . . . 8 ⊢ (ℕ0 ↑m 1o) = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
5	4	feq2i 6698	. . . . . . 7 ⊢ (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}):(ℕ0 ↑m 1o)⟶𝐾 ↔ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}):{ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
6	3, 5	sylibr 237	. . . . . 6 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}):(ℕ0 ↑m 1o)⟶𝐾)
7		rhmply1vsca.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		rhmply1vsca.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
9		rhmply1vsca.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑃)
10		rhmply1vsca.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅)
11	8, 9, 10	ply1basf 22331	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋:(ℕ0 ↑m 1o)⟶𝐾)
12	7, 11	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑋:(ℕ0 ↑m 1o)⟶𝐾)
13		rhmply1vsca.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
14		eqid 2769	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
15	10, 14	rhmf 20566	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:𝐾⟶(Base‘𝑆))
16	13, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐾⟶(Base‘𝑆))
17	16	ffnd 6707	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn 𝐾)
18		ovexd 7446	. . . . . 6 ⊢ (𝜑 → (ℕ0 ↑m 1o) ∈ V)
19		rhmrcl1 20558	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
20	13, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
21		eqid 2769	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
22	10, 21	ringcl 20332	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
23	20, 22	syl3an1 1179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
24	23	3expb 1136	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾)) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
25		eqid 2769	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
26	10, 21, 25	rhmmul 20568	. . . . . . . 8 ⊢ ((𝐻 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
27	13, 26	syl3an1 1179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
28	27	3expb 1136	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾)) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
29	6, 12, 17, 18, 24, 28	coof 7699	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋)) = ((𝐻 ∘ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶})) ∘f (.r‘𝑆)(𝐻 ∘ 𝑋)))
30		fcoconst 7131	. . . . . . 7 ⊢ ((𝐻 Fn 𝐾 ∧ 𝐶 ∈ 𝐾) → (𝐻 ∘ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶})) = ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}))
31	17, 1, 30	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶})) = ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}))
32	31	oveq1d 7426	. . . . 5 ⊢ (𝜑 → ((𝐻 ∘ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶})) ∘f (.r‘𝑆)(𝐻 ∘ 𝑋)) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}) ∘f (.r‘𝑆)(𝐻 ∘ 𝑋)))
33	29, 32	eqtrd 2804	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋)) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}) ∘f (.r‘𝑆)(𝐻 ∘ 𝑋)))
34		eqid 2769	. . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
35		eqid 2769	. . . . . 6 ⊢ ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘(1o mPoly 𝑅))
36	8, 9	ply1bas 22324	. . . . . 6 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅))
37		eqid 2769	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
38	34, 35, 10, 36, 21, 37, 1, 7	mplvsca 22133	. . . . 5 ⊢ (𝜑 → (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋))
39	38	coeq2d 5849	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)) = (𝐻 ∘ (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋)))
40		eqid 2769	. . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆)
41		eqid 2769	. . . . 5 ⊢ ( ·𝑠 ‘(1o mPoly 𝑆)) = ( ·𝑠 ‘(1o mPoly 𝑆))
42		rhmply1vsca.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝑆)
43		eqid 2769	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄)
44	42, 43	ply1bas 22324	. . . . 5 ⊢ (Base‘𝑄) = (Base‘(1o mPoly 𝑆))
45	16, 1	ffvelcdmd 7081	. . . . 5 ⊢ (𝜑 → (𝐻‘𝐶) ∈ (Base‘𝑆))
46		rhmghm 20565	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
47		ghmmhm 19296	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 GrpHom 𝑆) → 𝐻 ∈ (𝑅 MndHom 𝑆))
48	13, 46, 47	3syl 19	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
49	8, 42, 9, 43, 48, 7	mhmcoply1 22511	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ (Base‘𝑄))
50	40, 41, 14, 44, 25, 37, 45, 49	mplvsca 22133	. . . 4 ⊢ (𝜑 → ((𝐻‘𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻 ∘ 𝑋)) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}) ∘f (.r‘𝑆)(𝐻 ∘ 𝑋)))
51	33, 39, 50	3eqtr4d 2814	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)) = ((𝐻‘𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻 ∘ 𝑋)))
52		rhmply1vsca.t	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑃)
53	8, 34, 52	ply1vsca 22353	. . . . 5 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅))
54	53	oveqi 7424	. . . 4 ⊢ (𝐶 · 𝑋) = (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)
55	54	coeq2i 5847	. . 3 ⊢ (𝐻 ∘ (𝐶 · 𝑋)) = (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋))
56		rhmply1vsca.u	. . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝑄)
57	42, 40, 56	ply1vsca 22353	. . . 4 ⊢ ∙ = ( ·𝑠 ‘(1o mPoly 𝑆))
58	57	oveqi 7424	. . 3 ⊢ ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)) = ((𝐻‘𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻 ∘ 𝑋))
59	51, 55, 58	3eqtr4g 2829	. 2 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
60		rhmply1vsca.f	. . 3 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
61		coeq2 5845	. . 3 ⊢ (𝑝 = (𝐶 · 𝑋) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝐶 · 𝑋)))
62	8, 9, 10, 52, 20, 1, 7	ply1vscl 22510	. . 3 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
63	13, 62	coexd 7928	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) ∈ V)
64	60, 61, 62, 63	fvmptd3 7014	. 2 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = (𝐻 ∘ (𝐶 · 𝑋)))
65		coeq2 5845	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
66	13, 7	coexd 7928	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
67	60, 65, 7, 66	fvmptd3 7014	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
68	67	oveq2d 7427	. 2 ⊢ (𝜑 → ((𝐻‘𝐶) ∙ (𝐹‘𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
69	59, 64, 68	3eqtr4d 2814	1 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐹‘𝑋)))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {crab 3422  Vcvv 3461  {csn 4592   ↦ cmpt 5194   × cxp 5660  ^◡ccnv 5661   “ cima 5665   ∘ ccom 5666   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∘_f cof 7673  1_oc1o 8446   ↑_m cmap 8824  Fincfn 8943  ℕcn 12233  ℕ₀cn0 12504  Basecbs 17269  ._rcmulr 17311   ·_𝑠 cvsca 17314   MndHom cmhm 18839   GrpHom cghm 19283  Ringcrg 20315   RingHom crh 20551   mPoly cmpl 22025  Poly₁cpl1 22306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-sup 9402  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  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 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-hom 17334  df-cco 17335  df-0g 17494  df-prds 17500  df-pws 17502  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-grp 19003  df-minusg 19004  df-sbg 19005  df-subg 19189  df-ghm 19284  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-ring 20317  df-rhm 20554  df-lmod 20961  df-lss 21031  df-psr 22028  df-mpl 22030  df-opsr 22032  df-psr1 22309  df-ply1 22311

This theorem is referenced by:  rhmply1mon  22515