Theorem rhmply1vsca 22408

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 6798	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐾 → ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}):{ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}):{ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
4		psr1baslem 22202	. . . . . . . 8 ⊢ (ℕ0 ↑m 1o) = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
5	4	feq2i 6729	. . . . . . 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 234	. . . . . 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 22220	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋:(ℕ0 ↑m 1o)⟶𝐾)
12	7, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋:(ℕ0 ↑m 1o)⟶𝐾)
13		rhmply1vsca.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
14		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
15	10, 14	rhmf 20502	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:𝐾⟶(Base‘𝑆))
16	13, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐾⟶(Base‘𝑆))
17	16	ffnd 6738	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn 𝐾)
18		ovexd 7466	. . . . . 6 ⊢ (𝜑 → (ℕ0 ↑m 1o) ∈ V)
19		rhmrcl1 20493	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
20	13, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
21		eqid 2735	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
22	10, 21	ringcl 20268	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
23	20, 22	syl3an1 1162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
24	23	3expb 1119	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾)) → (𝑎(.r‘𝑅)𝑏) ∈ 𝐾)
25		eqid 2735	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
26	10, 21, 25	rhmmul 20503	. . . . . . . 8 ⊢ ((𝐻 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
27	13, 26	syl3an1 1162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (𝐻‘(𝑎(.r‘𝑅)𝑏)) = ((𝐻‘𝑎)(.r‘𝑆)(𝐻‘𝑏)))
28	27	3expb 1119	. . . . . 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 7721	. . . . 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 7154	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶})) = ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}))
32	31	oveq1d 7446	. . . . 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 2775	. . . 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 2735	. . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
35		eqid 2735	. . . . . 6 ⊢ ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘(1o mPoly 𝑅))
36	8, 9	ply1bas 22212	. . . . . 6 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅))
37		eqid 2735	. . . . . 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 22053	. . . . 5 ⊢ (𝜑 → (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋))
39	38	coeq2d 5876	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)) = (𝐻 ∘ (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋)))
40		eqid 2735	. . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆)
41		eqid 2735	. . . . 5 ⊢ ( ·𝑠 ‘(1o mPoly 𝑆)) = ( ·𝑠 ‘(1o mPoly 𝑆))
42		rhmply1vsca.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝑆)
43		eqid 2735	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄)
44	42, 43	ply1bas 22212	. . . . 5 ⊢ (Base‘𝑄) = (Base‘(1o mPoly 𝑆))
45	16, 1	ffvelcdmd 7105	. . . . 5 ⊢ (𝜑 → (𝐻‘𝐶) ∈ (Base‘𝑆))
46		rhmghm 20501	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
47		ghmmhm 19257	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 GrpHom 𝑆) → 𝐻 ∈ (𝑅 MndHom 𝑆))
48	13, 46, 47	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
49	8, 42, 9, 43, 48, 7	mhmcoply1 22405	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ (Base‘𝑄))
50	40, 41, 14, 44, 25, 37, 45, 49	mplvsca 22053	. . . 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 2785	. . 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 22242	. . . . 5 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅))
54	53	oveqi 7444	. . . 4 ⊢ (𝐶 · 𝑋) = (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)
55	54	coeq2i 5874	. . 3 ⊢ (𝐻 ∘ (𝐶 · 𝑋)) = (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋))
56		rhmply1vsca.u	. . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝑄)
57	42, 40, 56	ply1vsca 22242	. . . 4 ⊢ ∙ = ( ·𝑠 ‘(1o mPoly 𝑆))
58	57	oveqi 7444	. . 3 ⊢ ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)) = ((𝐻‘𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻 ∘ 𝑋))
59	51, 55, 58	3eqtr4g 2800	. 2 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
60		rhmply1vsca.f	. . 3 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
61		coeq2 5872	. . 3 ⊢ (𝑝 = (𝐶 · 𝑋) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝐶 · 𝑋)))
62	8, 9, 10, 52, 20, 1, 7	ply1vscl 22404	. . 3 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
63	13, 62	coexd 7954	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) ∈ V)
64	60, 61, 62, 63	fvmptd3 7039	. 2 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = (𝐻 ∘ (𝐶 · 𝑋)))
65		coeq2 5872	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
66	13, 7	coexd 7954	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
67	60, 65, 7, 66	fvmptd3 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
68	67	oveq2d 7447	. 2 ⊢ (𝜑 → ((𝐻‘𝐶) ∙ (𝐹‘𝑋)) = ((𝐻‘𝐶) ∙ (𝐻 ∘ 𝑋)))
69	59, 64, 68	3eqtr4d 2785	1 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐹‘𝑋)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478  {csn 4631   ↦ cmpt 5231   × cxp 5687  ^◡ccnv 5688   “ cima 5692   ∘ ccom 5693   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∘_f cof 7695  1_oc1o 8498   ↑_m cmap 8865  Fincfn 8984  ℕcn 12264  ℕ₀cn0 12524  Basecbs 17245  ._rcmulr 17299   ·_𝑠 cvsca 17302   MndHom cmhm 18807   GrpHom cghm 19243  Ringcrg 20251   RingHom crh 20486   mPoly cmpl 21944  Poly₁cpl1 22194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-pws 17496  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-rhm 20489  df-lmod 20877  df-lss 20948  df-psr 21947  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-ply1 22199

This theorem is referenced by:  rhmply1mon  22409