Theorem ressply1evls1 33664

Description: Subring evaluation of a univariate polynomial is the same as the subring evaluation in the bigger ring. (Contributed by Thierry Arnoux, 14-Nov-2025.)

Hypotheses

Ref	Expression
ressply1evls1.1	⊢ 𝐺 = (𝐸 ↾s 𝑅)
ressply1evls1.2	⊢ 𝑂 = (𝐸 evalSub1 𝑆)
ressply1evls1.3	⊢ 𝑄 = (𝐺 evalSub1 𝑆)
ressply1evls1.4	⊢ 𝑃 = (Poly1‘𝐾)
ressply1evls1.5	⊢ 𝐾 = (𝐸 ↾s 𝑆)
ressply1evls1.6	⊢ 𝐵 = (Base‘𝑃)
ressply1evls1.7	⊢ (𝜑 → 𝐸 ∈ CRing)
ressply1evls1.8	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝐸))
ressply1evls1.9	⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐺))
ressply1evls1.10	⊢ (𝜑 → 𝐹 ∈ 𝐵)

Assertion

Ref	Expression
ressply1evls1	⊢ (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))

Proof of Theorem ressply1evls1

Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressply1evls1.8	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝐸))
2		eqid 2737	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
3	2	subrgss 20522	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ⊆ (Base‘𝐸))
4		ressply1evls1.1	. . . . . 6 ⊢ 𝐺 = (𝐸 ↾s 𝑅)
5	4, 2	ressbas2 17179	. . . . 5 ⊢ (𝑅 ⊆ (Base‘𝐸) → 𝑅 = (Base‘𝐺))
6	1, 3, 5	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 = (Base‘𝐺))
7	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (Base‘𝐸))
8	6, 7	eqsstrrd 3971	. . 3 ⊢ (𝜑 → (Base‘𝐺) ⊆ (Base‘𝐸))
9	8	resmptd 6009	. 2 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐸) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))) ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
10		ressply1evls1.2	. . . 4 ⊢ 𝑂 = (𝐸 evalSub1 𝑆)
11		ressply1evls1.4	. . . 4 ⊢ 𝑃 = (Poly1‘𝐾)
12		ressply1evls1.5	. . . 4 ⊢ 𝐾 = (𝐸 ↾s 𝑆)
13		ressply1evls1.6	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
14		ressply1evls1.7	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
15		ressply1evls1.9	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐺))
16	4	subsubrg 20548	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (𝑆 ∈ (SubRing‘𝐺) ↔ (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅)))
17	16	biimpa 476	. . . . . 6 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ∈ (SubRing‘𝐺)) → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
18	1, 15, 17	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
19	18	simpld 494	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐸))
20		ressply1evls1.10	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
21		eqid 2737	. . . 4 ⊢ (.r‘𝐸) = (.r‘𝐸)
22		eqid 2737	. . . 4 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
23		eqid 2737	. . . 4 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
24	10, 2, 11, 12, 13, 14, 19, 20, 21, 22, 23	evls1fpws 22330	. . 3 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑥 ∈ (Base‘𝐸) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
25	24, 6	reseq12d 5949	. 2 ⊢ (𝜑 → ((𝑂‘𝐹) ↾ 𝑅) = ((𝑥 ∈ (Base‘𝐸) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))) ↾ (Base‘𝐺)))
26		ressply1evls1.3	. . . 4 ⊢ 𝑄 = (𝐺 evalSub1 𝑆)
27		eqid 2737	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
28		eqid 2737	. . . 4 ⊢ (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘(𝐺 ↾s 𝑆))
29		eqid 2737	. . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆)
30		eqid 2737	. . . 4 ⊢ (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘(Poly1‘(𝐺 ↾s 𝑆)))
31	4	subrgcrng 20525	. . . . 5 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → 𝐺 ∈ CRing)
32	14, 1, 31	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝐺 ∈ CRing)
33	18	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ 𝑅)
34		ressabs 17189	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅) → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
35	1, 33, 34	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
36	4	oveq1i 7380	. . . . . . . . . 10 ⊢ (𝐺 ↾s 𝑆) = ((𝐸 ↾s 𝑅) ↾s 𝑆)
37	35, 36, 12	3eqtr4g 2797	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾s 𝑆) = 𝐾)
38	37	fveq2d 6848	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘𝐾))
39	38, 11	eqtr4di 2790	. . . . . . 7 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = 𝑃)
40	39	fveq2d 6848	. . . . . 6 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘𝑃))
41	40, 13	eqtr4di 2790	. . . . 5 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = 𝐵)
42	20, 41	eleqtrrd 2840	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Poly1‘(𝐺 ↾s 𝑆))))
43		eqid 2737	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
44		eqid 2737	. . . 4 ⊢ (.g‘(mulGrp‘𝐺)) = (.g‘(mulGrp‘𝐺))
45	26, 27, 28, 29, 30, 32, 15, 42, 43, 44, 23	evls1fpws 22330	. . 3 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))))))
46		eqid 2737	. . . . . 6 ⊢ (+g‘𝐸) = (+g‘𝐸)
47	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐸 ∈ CRing)
48		nn0ex 12421	. . . . . . 7 ⊢ ℕ0 ∈ V
49	48	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ℕ0 ∈ V)
50	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑅 ⊆ (Base‘𝐸))
51	1	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ (SubRing‘𝐸))
52	33, 7	sstrd 3946	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
53	12, 2	ressbas2 17179	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (Base‘𝐸) → 𝑆 = (Base‘𝐾))
54	52, 53	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (Base‘𝐾))
55	54, 33	eqsstrrd 3971	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐾) ⊆ 𝑅)
56	55	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (Base‘𝐾) ⊆ 𝑅)
57	20	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝐹 ∈ 𝐵)
58		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
59		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	23, 13, 11, 59	coe1fvalcl 22170	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
61	57, 58, 60	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
62	56, 61	sseldd 3936	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ 𝑅)
63		eqid 2737	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
64	4, 63	mgpress 20102	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
65	47, 51, 64	syl2an2r 686	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
66	14	crngringd 20198	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
67		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (1r‘𝐸) = (1r‘𝐸)
68	67	subrg1cl 20530	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝑅)
69	1, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝑅)
70	4, 2, 67	ress1r 33333	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ 𝑅 ∧ 𝑅 ⊆ (Base‘𝐸)) → (1r‘𝐸) = (1r‘𝐺))
71	66, 69, 7, 70	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐺))
72	71	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (1r‘𝐸) = (1r‘𝐺))
73	63, 67	ringidval 20135	. . . . . . . . . . . 12 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
74		eqid 2737	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐺) = (mulGrp‘𝐺)
75		eqid 2737	. . . . . . . . . . . . 13 ⊢ (1r‘𝐺) = (1r‘𝐺)
76	74, 75	ringidval 20135	. . . . . . . . . . . 12 ⊢ (1r‘𝐺) = (0g‘(mulGrp‘𝐺))
77	72, 73, 76	3eqtr3g 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐺)))
78	63, 2	mgpbas 20097	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘(mulGrp‘𝐸))
79	7, 78	sseqtrdi 3976	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
80	79	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
81	6	eleq2d 2823	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ (Base‘𝐺)))
82	81	biimpar 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝑅)
83	82	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ 𝑅)
84	65, 77, 80, 58, 83	ressmulgnn0d 33144	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝑥))
85	74, 27	mgpbas 20097	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘(mulGrp‘𝐺))
86	4	subrgring 20524	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝐺 ∈ Ring)
87	74	ringmgp 20191	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (mulGrp‘𝐺) ∈ Mnd)
88	1, 86, 87	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝐺) ∈ Mnd)
89	88	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝐺) ∈ Mnd)
90		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ (Base‘𝐺))
91	85, 44, 89, 58, 90	mulgnn0cld 19042	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) ∈ (Base‘𝐺))
92	84, 91	eqeltrrd 2838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝑥) ∈ (Base‘𝐺))
93	51, 3, 5	3syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 = (Base‘𝐺))
94	92, 93	eleqtrrd 2840	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝑥) ∈ 𝑅)
95	21, 51, 62, 94	subrgmcld 33332	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)) ∈ 𝑅)
96	95	fmpttd 7071	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))):ℕ0⟶𝑅)
97		subrgsubg 20527	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ∈ (SubGrp‘𝐸))
98		eqid 2737	. . . . . . . . 9 ⊢ (0g‘𝐸) = (0g‘𝐸)
99	98	subg0cl 19081	. . . . . . . 8 ⊢ (𝑅 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝑅)
100	1, 97, 99	3syl 18	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝑅)
101	100	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (0g‘𝐸) ∈ 𝑅)
102	14	crnggrpd 20199	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
103	102	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝐸 ∈ Grp)
104		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝑦 ∈ (Base‘𝐸))
105	2, 46, 98, 103, 104	grplidd 18916	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → ((0g‘𝐸)(+g‘𝐸)𝑦) = 𝑦)
106	2, 46, 98, 103, 104	grpridd 18917	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → (𝑦(+g‘𝐸)(0g‘𝐸)) = 𝑦)
107	105, 106	jca 511	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → (((0g‘𝐸)(+g‘𝐸)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐸)(0g‘𝐸)) = 𝑦))
108	2, 46, 4, 47, 49, 50, 96, 101, 107	gsumress 18621	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))) = (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))))
109	4, 21	ressmulr 17241	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐺))
110	1, 109	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐺))
111	110	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (.r‘𝐸) = (.r‘𝐺))
112	111	oveqd 7387	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐺))𝑥)) = (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)))
113	84	oveq2d 7386	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐺))𝑥)) = (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))
114	112, 113	eqtr3d 2774	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)) = (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))
115	114	mpteq2dva 5193	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))) = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))
116	115	oveq2d 7386	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)))) = (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))))
117	108, 116	eqtr4d 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))) = (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)))))
118	117	mpteq2dva 5193	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐺) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))))))
119	45, 118	eqtr4d 2775	. 2 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
120	9, 25, 119	3eqtr4rd 2783	1 ⊢ (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903   ↦ cmpt 5181   ↾ cres 5636  ‘cfv 6502  (class class class)co 7370  ℕ₀cn0 12415  Basecbs 17150   ↾_s cress 17171  +_gcplusg 17191  ._rcmulr 17192  0_gc0g 17373   Σ_g cgsu 17374  Mndcmnd 18673  Grpcgrp 18880  ._gcmg 19014  SubGrpcsubg 19067  mulGrpcmgp 20092  1_rcur 20133  Ringcrg 20185  CRingccrg 20186  SubRingcsubrg 20519  Poly₁cpl1 22134  coe₁cco1 22135   evalSub₁ ces1 22274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-ofr 7635  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-pm 8780  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-sup 9359  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-fzo 13585  df-seq 13939  df-hash 14268  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-hom 17215  df-cco 17216  df-0g 17375  df-gsum 17376  df-prds 17381  df-pws 17383  df-mre 17519  df-mrc 17520  df-acs 17522  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-mulg 19015  df-subg 19070  df-ghm 19159  df-cntz 19263  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-srg 20139  df-ring 20187  df-cring 20188  df-rhm 20425  df-subrng 20496  df-subrg 20520  df-lmod 20830  df-lss 20900  df-lsp 20940  df-assa 21825  df-asp 21826  df-ascl 21827  df-psr 21882  df-mvr 21883  df-mpl 21884  df-opsr 21886  df-evls 22046  df-evl 22047  df-psr1 22137  df-vr1 22138  df-ply1 22139  df-coe1 22140  df-evls1 22276  df-evl1 22277

This theorem is referenced by:  cos9thpiminply  33972