Theorem rhmply1vsca 22611

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 22411	. . . . . . . 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 22428	. . . . . . 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 20627	. . . . . . . 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 7451	. . . . . 6 ⊢ (𝜑 → (ℕ0 ↑m 1o) ∈ V)
19		rhmrcl1 20619	. . . . . . . . 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 20390	. . . . . . . 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 2762	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
26	10, 21, 25	rhmmul 20632	. . . . . . . 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 7705	. . . . 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 596	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶})) = ({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(𝐻‘𝐶)}))
32	31	oveq1d 7431	. . . . 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 22421	. . . . . 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 22230	. . . . 5 ⊢ (𝜑 → (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋) = (({ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r‘𝑅)𝑋))
39	38	coeq2d 5846	. . . 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 22421	. . . . 5 ⊢ (Base‘𝑄) = (Base‘(1o mPoly 𝑆))
45	16, 1	ffvelcdmd 7081	. . . . 5 ⊢ (𝜑 → (𝐻‘𝐶) ∈ (Base‘𝑆))
46		rhmghm 20626	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
47		ghmmhm 19354	. . . . . . 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 22608	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ (Base‘𝑄))
50	40, 41, 14, 44, 25, 37, 45, 49	mplvsca 22230	. . . 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 22450	. . . . 5 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅))
54	53	oveqi 7429	. . . 4 ⊢ (𝐶 · 𝑋) = (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)
55	54	coeq2i 5844	. . 3 ⊢ (𝐻 ∘ (𝐶 · 𝑋)) = (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋))
56		rhmply1vsca.u	. . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝑄)
57	42, 40, 56	ply1vsca 22450	. . . 4 ⊢ ∙ = ( ·𝑠 ‘(1o mPoly 𝑆))
58	57	oveqi 7429	. . 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 5842	. . 3 ⊢ (𝑝 = (𝐶 · 𝑋) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝐶 · 𝑋)))
62	8, 9, 10, 52, 20, 1, 7	ply1vscl 22607	. . 3 ⊢ (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
63	13, 62	coexd 7931	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) ∈ V)
64	60, 61, 62, 63	fvmptd3 7014	. 2 ⊢ (𝜑 → (𝐹‘(𝐶 · 𝑋)) = (𝐻 ∘ (𝐶 · 𝑋)))
65		coeq2 5842	. . . 4 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
66	13, 7	coexd 7931	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
67	60, 65, 7, 66	fvmptd3 7014	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
68	67	oveq2d 7432	. 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 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  {csn 4587   ↦ cmpt 5190   × cxp 5657  ^◡ccnv 5658   “ cima 5662   ∘ ccom 5663   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∘_f cof 7679  1_oc1o 8451   ↑_m cmap 8829  Fincfn 8955  ℕcn 12260  ℕ₀cn0 12531  Basecbs 17305  ._rcmulr 17347   ·_𝑠 cvsca 17350   MndHom cmhm 18890   GrpHom cghm 19341  Ringcrg 20373   RingHom crh 20611   mPoly cmpl 22122  Poly₁cpl1 22403

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2100  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 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-sup 9415  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13564  df-struct 17243  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-sca 17362  df-vsca 17363  df-ip 17364  df-tset 17365  df-ple 17366  df-ds 17368  df-hom 17370  df-cco 17371  df-0g 17530  df-prds 17536  df-pws 17538  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-grp 19061  df-minusg 19062  df-sbg 19063  df-subg 19247  df-ghm 19342  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-rhm 20614  df-lmod 21047  df-lss 21117  df-psr 22125  df-mpl 22127  df-opsr 22129  df-psr1 22406  df-ply1 22408

This theorem is used by:  rhmply1mon  22612