Theorem rhmply1vsca 22730

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 6790	. . . . . . . 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 22530	. . . . . . . 8 ⊢ (ℕ0 ↑m 1o) = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
5	4	feq2i 6720	. . . . . . 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 22547	. . . . . . 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 2763	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
15	10, 14	rhmf 20742	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:𝐾⟶(Base‘𝑆))
16	13, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐾⟶(Base‘𝑆))
17	16	ffnd 6729	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn 𝐾)
18		ovexd 7475	. . . . . 6 ⊢ (𝜑 → (ℕ0 ↑m 1o) ∈ V)
19		rhmrcl1 20734	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
20	13, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
21		eqid 2763	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
22	10, 21	ringcl 20504	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
23	20, 22	syl3an1 1181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
24	23	3expb 1138	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾)) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
25		eqid 2763	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
26	10, 21, 25	rhmmul 20747	. . . . . . . 8 ⊢ ((𝐻 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
27	13, 26	syl3an1 1181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
28	27	3expb 1138	. . . . . 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 7738	. . . . 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 7156	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶})) = ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}))
32	31	oveq1d 7455	. . . . 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 2798	. . . 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 2763	. . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
35		eqid 2763	. . . . . 6 ⊢ ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘(1o mPoly 𝑅))
36	8, 9	ply1bas 22540	. . . . . 6 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅))
37		eqid 2763	. . . . . 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 22349	. . . . 5 ⊢ (𝜑 → (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋))
39	38	coeq2d 5845	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)) = (𝐻 ∘ (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋)))
40		eqid 2763	. . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆)
41		eqid 2763	. . . . 5 ⊢ ( ·𝑠 ‘(1o mPoly 𝑆)) = ( ·𝑠 ‘(1o mPoly 𝑆))
42		rhmply1vsca.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝑆)
43		eqid 2763	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄)
44	42, 43	ply1bas 22540	. . . . 5 ⊢ (Base‘𝑄) = (Base‘(1o mPoly 𝑆))
45	16, 1	ffvelcdmd 7106	. . . . 5 ⊢ (𝜑 → (𝐻‘𝐶) ∈ (Base‘𝑆))
46		rhmghm 20741	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
47		ghmmhm 19466	. . . . . . 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 22727	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ (Base‘𝑄))
50	40, 41, 14, 44, 25, 37, 45, 49	mplvsca 22349	. . . 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 2808	. . 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 22569	. . . . 5 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅))
54	53	oveqi 7453	. . . 4 ⊢ (𝐶 · 𝑋) = (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)
55	54	coeq2i 5843	. . 3 ⊢ (𝐻 ∘ (𝐶 · 𝑋)) = (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋))
56		rhmply1vsca.u	. . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝑄)
57	42, 40, 56	ply1vsca 22569	. . . 4 ⊢ ∙ = ( ·𝑠 ‘(1o mPoly 𝑆))
58	57	oveqi 7453	. . 3 ⊢ ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)) = ((𝐻‘𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻 ∘ 𝑋))
59	51, 55, 58	3eqtr4g 2823	. 2 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
60		rhmply1vsca.f	. . 3 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
61		coeq2 5841	. . 3 ⊢ (𝑝 = (𝐶 · 𝑋) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝐶 · 𝑋)))
62	8, 9, 10, 52, 20, 1, 7	ply1vscl 22726	. . 3 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
63	13, 62	coexd 7964	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) ∈ V)
64	60, 61, 62, 63	fvmptd3 7037	. 2 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = (𝐻 ∘ (𝐶 · 𝑋)))
65		coeq2 5841	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
66	13, 7	coexd 7964	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
67	60, 65, 7, 66	fvmptd3 7037	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
68	67	oveq2d 7456	. 2 ⊢ (𝜑 → ((𝐻‘𝐶) ∙ (𝐹‘𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
69	59, 64, 68	3eqtr4d 2808	1 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐹‘𝑋)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  {csn 4588   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654   “ cima 5658   ∘ ccom 5659   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ∘_f cof 7712  1_oc1o 8494   ↑_m cmap 8872  Fincfn 8998  ℕcn 12359  ℕ₀cn0 12630  Basecbs 17413  ._rcmulr 17455   ·_𝑠 cvsca 17458   MndHom cmhm 19002   GrpHom cghm 19453  Ringcrg 20486   RingHom crh 20726   mPoly cmpl 22241  Poly₁cpl1 22522

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-map 8874  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-sup 9458  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-hom 17478  df-cco 17479  df-0g 17638  df-prds 17644  df-pws 17646  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-grp 19173  df-minusg 19174  df-sbg 19175  df-subg 19359  df-ghm 19454  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-rhm 20729  df-lmod 21164  df-lss 21234  df-psr 22244  df-mpl 22246  df-opsr 22248  df-psr1 22525  df-ply1 22527

This theorem is used by:  rhmply1mon  22731