Theorem ressply1evls1 33962

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 2762	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
3	2	subrgss 20735	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ⊆ (Base‘𝐸))
4		ressply1evls1.1	. . . . . 6 ⊢ 𝐺 = (𝐸 ↾s 𝑅)
5	4, 2	ressbas2 17334	. . . . 5 ⊢ (𝑅 ⊆ (Base‘𝐸) → 𝑅 = (Base‘𝐺))
6	1, 3, 5	3syl 19	. . . 4 ⊢ (𝜑 → 𝑅 = (Base‘𝐺))
7	1, 3	syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (Base‘𝐸))
8	6, 7	eqsstrrd 3969	. . 3 ⊢ (𝜑 → (Base‘𝐺) ⊆ (Base‘𝐸))
9	8	resmptd 6040	. 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 20761	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (𝑆 ∈ (SubRing‘𝐺) ↔ (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅)))
17	16	biimpa 482	. . . . . 6 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ∈ (SubRing‘𝐺)) → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
18	1, 15, 17	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
19	18	simpld 500	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐸))
20		ressply1evls1.10	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
21		eqid 2762	. . . 4 ⊢ (.r‘𝐸) = (.r‘𝐸)
22		eqid 2762	. . . 4 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
23		eqid 2762	. . . 4 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
24	10, 2, 11, 12, 13, 14, 19, 20, 21, 22, 23	evls1fpws 22595	. . 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 5977	. 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 2762	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
28		eqid 2762	. . . 4 ⊢ (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘(𝐺 ↾s 𝑆))
29		eqid 2762	. . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆)
30		eqid 2762	. . . 4 ⊢ (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘(Poly1‘(𝐺 ↾s 𝑆)))
31	4	subrgcrng 20738	. . . . 5 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → 𝐺 ∈ CRing)
32	14, 1, 31	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝐺 ∈ CRing)
33	18	simprd 501	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ 𝑅)
34		ressabs 17344	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅) → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
35	1, 33, 34	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
36	4	oveq1i 7426	. . . . . . . . . 10 ⊢ (𝐺 ↾s 𝑆) = ((𝐸 ↾s 𝑅) ↾s 𝑆)
37	35, 36, 12	3eqtr4g 2822	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾s 𝑆) = 𝐾)
38	37	fveq2d 6886	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘𝐾))
39	38, 11	eqtr4di 2815	. . . . . . 7 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = 𝑃)
40	39	fveq2d 6886	. . . . . 6 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘𝑃))
41	40, 13	eqtr4di 2815	. . . . 5 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = 𝐵)
42	20, 41	eleqtrrd 2865	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Poly1‘(𝐺 ↾s 𝑆))))
43		eqid 2762	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
44		eqid 2762	. . . 4 ⊢ (.g‘(mulGrp‘𝐺)) = (.g‘(mulGrp‘𝐺))
45	26, 27, 28, 29, 30, 32, 15, 42, 43, 44, 23	evls1fpws 22595	. . 3 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))))))
46		eqid 2762	. . . . . 6 ⊢ (+g‘𝐸) = (+g‘𝐸)
47	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐸 ∈ CRing)
48		nn0ex 12537	. . . . . . 7 ⊢ ℕ0 ∈ V
49	48	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ℕ0 ∈ V)
50	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑅 ⊆ (Base‘𝐸))
51	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ (SubRing‘𝐸))
52	33, 7	sstrd 3944	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
53	12, 2	ressbas2 17334	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (Base‘𝐸) → 𝑆 = (Base‘𝐾))
54	52, 53	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (Base‘𝐾))
55	54, 33	eqsstrrd 3969	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐾) ⊆ 𝑅)
56	55	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (Base‘𝐾) ⊆ 𝑅)
57	20	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝐹 ∈ 𝐵)
58		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
59		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	23, 13, 11, 59	coe1fvalcl 22438	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
61	57, 58, 60	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
62	56, 61	sseldd 3935	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ 𝑅)
63		eqid 2762	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
64	4, 63	mgpress 20284	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
65	47, 51, 64	syl2an2r 698	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
66	14	crngringd 20386	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
67		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (1r‘𝐸) = (1r‘𝐸)
68	67	subrg1cl 20743	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝑅)
69	1, 68	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝑅)
70	4, 2, 67	ress1r 33659	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ 𝑅 ∧ 𝑅 ⊆ (Base‘𝐸)) → (1r‘𝐸) = (1r‘𝐺))
71	66, 69, 7, 70	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐺))
72	71	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (1r‘𝐸) = (1r‘𝐺))
73	63, 67	ringidval 20323	. . . . . . . . . . . 12 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
74		eqid 2762	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐺) = (mulGrp‘𝐺)
75		eqid 2762	. . . . . . . . . . . . 13 ⊢ (1r‘𝐺) = (1r‘𝐺)
76	74, 75	ringidval 20323	. . . . . . . . . . . 12 ⊢ (1r‘𝐺) = (0g‘(mulGrp‘𝐺))
77	72, 73, 76	3eqtr3g 2820	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐺)))
78	63, 2	mgpbas 20279	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘(mulGrp‘𝐸))
79	7, 78	sseqtrdi 3974	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
80	79	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
81	6	eleq2d 2848	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ (Base‘𝐺)))
82	81	biimpar 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝑅)
83	82	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ 𝑅)
84	65, 77, 80, 58, 83	ressmulgnn0d 33471	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝑥))
85	74, 27	mgpbas 20279	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘(mulGrp‘𝐺))
86	4	subrgring 20737	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝐺 ∈ Ring)
87	74	ringmgp 20379	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (mulGrp‘𝐺) ∈ Mnd)
88	1, 86, 87	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝐺) ∈ Mnd)
89	88	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝐺) ∈ Mnd)
90		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ (Base‘𝐺))
91	85, 44, 89, 58, 90	mulgnn0cld 19219	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) ∈ (Base‘𝐺))
92	84, 91	eqeltrrd 2863	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝑥) ∈ (Base‘𝐺))
93	51, 3, 5	3syl 19	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 = (Base‘𝐺))
94	92, 93	eleqtrrd 2865	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝑥) ∈ 𝑅)
95	21, 51, 62, 94	subrgmcld 33658	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)) ∈ 𝑅)
96	95	fmpttd 7111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))):ℕ0⟶𝑅)
97		subrgsubg 20740	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ∈ (SubGrp‘𝐸))
98		eqid 2762	. . . . . . . . 9 ⊢ (0g‘𝐸) = (0g‘𝐸)
99	98	subg0cl 19258	. . . . . . . 8 ⊢ (𝑅 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝑅)
100	1, 97, 99	3syl 19	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝑅)
101	100	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (0g‘𝐸) ∈ 𝑅)
102	14	crnggrpd 20387	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
103	102	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝐸 ∈ Grp)
104		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝑦 ∈ (Base‘𝐸))
105	2, 46, 98, 103, 104	grplidd 19094	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → ((0g‘𝐸)(+g‘𝐸)𝑦) = 𝑦)
106	2, 46, 98, 103, 104	grpridd 19095	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → (𝑦(+g‘𝐸)(0g‘𝐸)) = 𝑦)
107	105, 106	jca 521	. . . . . 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 18786	. . . . 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 17396	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐺))
110	1, 109	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐺))
111	110	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (.r‘𝐸) = (.r‘𝐺))
112	111	oveqd 7433	. . . . . . . 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 7432	. . . . . . . 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 2799	. . . . . . 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 5202	. . . . . 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 7432	. . . . 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 2800	. . . 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 5202	. . 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 2800	. 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 2808	1 ⊢ (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902   ↦ cmpt 5190   ↾ cres 5661  ‘cfv 6537  (class class class)co 7416  ℕ₀cn0 12531  Basecbs 17305   ↾_s cress 17326  +_gcplusg 17346  ._rcmulr 17347  0_gc0g 17528   Σ_g cgsu 17529  Mndcmnd 18838  Grpcgrp 19058  ._gcmg 19191  SubGrpcsubg 19244  mulGrpcmgp 20274  1_rcur 20321  Ringcrg 20373  CRingccrg 20374  SubRingcsubrg 20732  Poly₁cpl1 22403  coe₁cco1 22404   evalSub₁ ces1 22539

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-int 4911  df-iun 4956  df-iin 4957  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-se 5613  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-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-ofr 7682  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-2o 8459  df-er 8699  df-map 8831  df-pm 8832  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-sup 9415  df-oi 9485  df-card 9947  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-fzo 13712  df-seq 14068  df-hash 14397  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-gsum 17531  df-prds 17536  df-pws 17538  df-mre 17674  df-mrc 17675  df-acs 17677  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-submnd 18893  df-grp 19061  df-minusg 19062  df-sbg 19063  df-mulg 19192  df-subg 19247  df-ghm 19342  df-cntz 19445  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-srg 20327  df-ring 20375  df-cring 20376  df-rhm 20614  df-subrng 20709  df-subrg 20733  df-lmod 21047  df-lss 21117  df-lsp 21157  df-assa 22069  df-asp 22070  df-ascl 22071  df-psr 22125  df-mvr 22126  df-mpl 22127  df-opsr 22129  df-evls 22291  df-evl 22292  df-psr1 22406  df-vr1 22407  df-ply1 22408  df-coe1 22409  df-evls1 22541  df-evl1 22542

This theorem is used by:  cos9thpiminply  34285