Theorem rhmply1vsca 22556

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 6767	. . . . . . . 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 22356	. . . . . . . 8 ⊢ (ℕ0 ↑m 1o) = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
5	4	feq2i 6697	. . . . . . 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 22373	. . . . . . 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 2762	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
15	10, 14	rhmf 20574	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:𝐾⟶(Base‘𝑆))
16	13, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐾⟶(Base‘𝑆))
17	16	ffnd 6706	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn 𝐾)
18		ovexd 7447	. . . . . 6 ⊢ (𝜑 → (ℕ0 ↑m 1o) ∈ V)
19		rhmrcl1 20566	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
20	13, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
21		eqid 2762	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
22	10, 21	ringcl 20338	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
23	20, 22	syl3an1 1180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
24	23	3expb 1137	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾)) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
25		eqid 2762	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
26	10, 21, 25	rhmmul 20579	. . . . . . . 8 ⊢ ((𝐻 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
27	13, 26	syl3an1 1180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
28	27	3expb 1137	. . . . . 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 7700	. . . . 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 7130	. . . . . . 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 7427	. . . . 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 2797	. . . 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 2762	. . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
35		eqid 2762	. . . . . 6 ⊢ ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘(1o mPoly 𝑅))
36	8, 9	ply1bas 22366	. . . . . 6 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅))
37		eqid 2762	. . . . . 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 22175	. . . . 5 ⊢ (𝜑 → (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋))
39	38	coeq2d 5847	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)) = (𝐻 ∘ (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋)))
40		eqid 2762	. . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆)
41		eqid 2762	. . . . 5 ⊢ ( ·𝑠 ‘(1o mPoly 𝑆)) = ( ·𝑠 ‘(1o mPoly 𝑆))
42		rhmply1vsca.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝑆)
43		eqid 2762	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄)
44	42, 43	ply1bas 22366	. . . . 5 ⊢ (Base‘𝑄) = (Base‘(1o mPoly 𝑆))
45	16, 1	ffvelcdmd 7080	. . . . 5 ⊢ (𝜑 → (𝐻‘𝐶) ∈ (Base‘𝑆))
46		rhmghm 20573	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
47		ghmmhm 19302	. . . . . . 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 22553	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ (Base‘𝑄))
50	40, 41, 14, 44, 25, 37, 45, 49	mplvsca 22175	. . . 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 2807	. . 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 22395	. . . . 5 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅))
54	53	oveqi 7425	. . . 4 ⊢ (𝐶 · 𝑋) = (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)
55	54	coeq2i 5845	. . 3 ⊢ (𝐻 ∘ (𝐶 · 𝑋)) = (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋))
56		rhmply1vsca.u	. . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝑄)
57	42, 40, 56	ply1vsca 22395	. . . 4 ⊢ ∙ = ( ·𝑠 ‘(1o mPoly 𝑆))
58	57	oveqi 7425	. . 3 ⊢ ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)) = ((𝐻‘𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻 ∘ 𝑋))
59	51, 55, 58	3eqtr4g 2822	. 2 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
60		rhmply1vsca.f	. . 3 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
61		coeq2 5843	. . 3 ⊢ (𝑝 = (𝐶 · 𝑋) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝐶 · 𝑋)))
62	8, 9, 10, 52, 20, 1, 7	ply1vscl 22552	. . 3 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
63	13, 62	coexd 7926	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) ∈ V)
64	60, 61, 62, 63	fvmptd3 7013	. 2 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = (𝐻 ∘ (𝐶 · 𝑋)))
65		coeq2 5843	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
66	13, 7	coexd 7926	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
67	60, 65, 7, 66	fvmptd3 7013	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
68	67	oveq2d 7428	. 2 ⊢ (𝜑 → ((𝐻‘𝐶) ∙ (𝐹‘𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
69	59, 64, 68	3eqtr4d 2807	1 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐹‘𝑋)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {crab 3415  Vcvv 3454  {csn 4588   ↦ cmpt 5191   × cxp 5658  ^◡ccnv 5659   “ cima 5663   ∘ ccom 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ∘_f cof 7674  1_oc1o 8444   ↑_m cmap 8822  Fincfn 8941  ℕcn 12239  ℕ₀cn0 12510  Basecbs 17275  ._rcmulr 17317   ·_𝑠 cvsca 17320   MndHom cmhm 18845   GrpHom cghm 19289  Ringcrg 20321   RingHom crh 20558   mPoly cmpl 22067  Poly₁cpl1 22348

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8155  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9320  df-sup 9400  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-fz 13542  df-struct 17213  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-mulr 17330  df-sca 17332  df-vsca 17333  df-ip 17334  df-tset 17335  df-ple 17336  df-ds 17338  df-hom 17340  df-cco 17341  df-0g 17500  df-prds 17506  df-pws 17508  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-mhm 18847  df-grp 19009  df-minusg 19010  df-sbg 19011  df-subg 19195  df-ghm 19290  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-rhm 20561  df-lmod 20994  df-lss 21064  df-psr 22070  df-mpl 22072  df-opsr 22074  df-psr1 22351  df-ply1 22353

This theorem is used by:  rhmply1mon  22557